- + 1.2 Hamilton’s variational principle
- + 1.3 First steps with conserved quantities
- + 1.4 Euler-Lagrange equations on smooth manifolds
- + 2.2 The spherical pendulum
- + 2.3 Intermezzo: small oscillations
- + 2.4 Motion in a central potential
- + 2.5 D’Alembert principle and systems with constraints
- + 3.1 The Legendre Transform in the euclidean plane
- + 3.2 Intermezzo: cotangent bundle and differential forms
- + 3.3 The Legendre Transform
- + 3.4 Poisson brackets and first integrals of motion
- + 3.5 Variational principles of hamiltonian mechanics
- + 3.6 Canonical transformations
- + 3.7 The symplectic structure on the cotangent bundle
- + 4.1 Lagrangian submanifolds
- + 4.2 Canonical transformations revisited
- + 4.3 Hamilton-Jacobi equations
- + 4.4 The Liouville-Arnold theorem
- + 5 Hamiltonian perturbation theory
- + 5.1 Small oscillations revisited
- + 5.3 A brief look at KAM theory
++ +Index +
+ + @@ -659,9 +665,10 @@-In the elementary formulation we will focus on, our primary interest is in describing the equations of motion for an idealized point particle. This is a point–like object obtained by ignoring the dimensions of the physical object. Note that in -many cases this is a reasonable first approximation. For example, when describing planetary motion around the sun, we can consider the planet and the sun as two point particles without significantly affecting the qualitative properties of the system. -After all the sun is about 150 million kilometers away from the Earth, compared to which their sizes are negligible. The diameter of the sun is just 0.9% of this distance and the diameter of the Earth a mere 0,008%. +In the elementary formulation we will focus on, our primary interest is in describing the equations of motion for an idealized point particle. This is a point–like object obtained by ignoring the dimensions of the +physical object. Note that in many cases this is a reasonable first approximation. For example, when describing planetary motion around the sun, we can consider the planet and the sun as two point particles without significantly affecting the +qualitative properties of the system. After all the sun is about 150 million kilometers away from the Earth, compared to which their sizes are negligible. The diameter of the sun is just 0.9% of this distance and the diameter of the Earth a mere +0,008%. See Figure 1.1.
-A point particle is first of all a point, a dimensionless mathematical object in space: its position in space is described by the position vector \(\vb *{x} = (x, y ,z)\). Keep in mind that +A point particle is first of all a point, a dimensionless mathematical object in space: its position in space is described by the position vector \(\vb *{x} = (x, y ,z)\). Keep in mind that
\(\seteqnumber{0}{}{0}\) @@ -718,7 +727,7 @@-The velocity of the point particle is given by the rate of change of the position vector, i.e., its derivative with respect to time1 +The velocity of the point particle is given by the rate of change of the position vector, i.e., its derivative with respect to time1
\(\seteqnumber{0}{}{1}\) @@ -739,7 +748,7 @@-We call acceleration, the rate of change of the velocity, i.e., the second derivative +We call acceleration, the rate of change of the velocity, i.e., the second derivative
\(\seteqnumber{0}{}{2}\) @@ -762,13 +771,13 @@For us, the main distinction between a mathematical point and a point particle, is that the latter usually carries a mass \(m\). This is a scalar quantity that, as we will see, measures its resistance to changes in its state of motion, also -known as inertia. +known as inertia.
-1 Notation: the symbol \(a := b\) means that \(a\) is defined by the expression \(b\), similarly \(b =: a\) is the same statement but read from right to left. +1 Notation: the symbol \(a := b\) means that \(a\) is defined by the expression \(b\). Similarly \(b =: a\) is the same statement but read from right to left.
@@ -781,14 +790,27 @@+Newton’s second law of motion +
+-The mechanics of the particle is encoded by Newton’s second law of motion. That is, there exist frames of reference (i.e systems of coordinates) in which the motion of the particle is described by a differential equation involving -the forces \(\vb *{F}\) acting on the point particle, its mass \(m\) and its acceleration as follows2 +There exist frames of reference, that is, systems of coordinates, in which the motion of the particle is described by a differential equation involving the forces \(\vb *{F}\) acting on the point particle, its mass +\(m\) and its acceleration as follows
\(\seteqnumber{0}{}{3}\) @@ -808,43 +830,18 @@-2 This is not completely true, Newton was already talking about momentum, which makes a deep difference but would set us off-course right now. -
- --Let’s leave it here for now, we will come back to it later on. +Note that the law as described above is not completely true: Newton was already talking about momentum, which makes a deep difference, but would set us off-course right now.
-In general we will consider systems of \(N\) point particles. These will be described by a set of \(N\) position vectors \(\vb *{x}_k = (x_k, y_k ,z_k)\) with masses \(m_k\), \(k = 1, \ldots , N\). For convenience we will denote3 -\(\vb *{x} = (\vb *{x}_1, \ldots , \vb *{x}_N)\in \mathbb {R}^{3N}\), and \(\vb *{m} = (m_1, \ldots , m_N)\in \mathbb {R}^N\). We call \(\vb *{x}(t)\) the configuration of the system at time \(t\) in the -configuration space \(\mathbb {R}^{3N}\). +Before moving on, let’s anticipate three examples that will come back over and over in the course of this book. Here we will just sketch them, but we will come back to them in more detail later.
--3 Here we are identifying \(\mathbb {R}^{3\times N}\), the space of \(3\times N\) matrices, and \(\mathbb {R}^{3N}\), the space of \(3N\) vectors. Namely, \(\vb *{x} = (\vb *{x}_1, \ldots , \vb *{x}_N) = (x_1, y_1, -z_1, x_2, y_2, z_2, \ldots , x_N, y_N, z_N)\) is used interchangeably as the vector of positions of the points in three-space or as the vector including all the positions of the \(N\) bodies together. -
- - -- -Example 1.1. For example a system of two rigid pendulums in space -constrained to oscillate on a vertical plane, is described by two position vectors, so \(\vb *{x} = (\vb *{x}_1, \vb *{x}_2)\in \mathbb {R}^{6}\). However, to describe their configuration we only need two angular variables, one for each of the -pendulums. So, for all practical purposes, the system could be completely described4 by \(q = (q^1, q^2) \in \mathbb {S}^1\times \mathbb {S}^1 \simeq \mathbb {T}^2\). See Figure 1.2. -
- --4 Here there is an hidden statement: \(\mathbb {S}^1\times \mathbb {S}^1 \simeq \mathbb {T}^2\). This should be read as: the 2-torus, the surface of a doughnut, is diffemorphic to the product of two circles. Here, -diffeomorphic is a concept from the theory of smooth manifolds, and stands for the existence a smooth bijection between the two constructions. Note that \(\simeq \) usually denotes some kind of equivalence, but which specific kind is -context-dependent and not often well explicited. + +Example 1.1 (Horizontal spring and pendulum). Consider an ideal spring, that is, an idealized system consisting of a point particle of mass \(m\) attached to a spring with stiffness \(k\), sliding on a frictionless surface. +Assume that the motion is one-dimensional, along the axis of the spring, and let \(x\) denote the displacement of the system from its equilibrium position, i.e., the position in which the spring is completely at rest: not compressed nor extended. It is +convenient to assume that the sliding happens on a flat surface. This allows us to exclude gravitational forces from the picture, making this example as simple as it could be.
- - --We say that a system of \(N\) particles has \(n\) degrees of freedom if we need \(n\) independent parameters to uniquely specify the system configuration5. We call generalized -coordinates any set of \(n\) parameters \(q = (q^1, \ldots , q^n)\) that uniquely determine the configuration of a system with \(n\) degrees of freedom, and generalized velocities their time derivatives \(\dot q = (\dot q^1, -\ldots , \dot q^n)\). The state of the system is then characterized by the set of (generalized) coordinates and velocities \((q, \dot q) = \left (q^1, \ldots , q^n,\dot q^1, \ldots , \dot q^n\right )\). -
- --If you recall differential geometry, you may (correctly) guess that the generalized coordinates will be points on some differentiable manifold \(q\in M\), their evolution will be described by a curve \(q: \mathbb {R} \to M\) parametrized by time and -the state of the system will be a point in the tangent bundle \((q, \dot q)\in TM\), i.e., \(\dot q \in T_q M\) (see also section 1.4). -
- --We have now all the elements to translate the Newtonian principle of determinacy in mathematical terms. In 1814, Laplace [Lap51] wrote -
-- -- --We may regard the present state of the universe as the effect of its past and the cause of its future. An intellect which at a certain moment would know all forces that set nature in motion, and all positions of all items of which nature is composed, if -this intellect were also vast enough to submit these data to analysis, it would embrace in a single formula the movements of the greatest bodies of the universe and those of the tiniest atom; for such an intellect nothing would be uncertain and the -future just like the past would be present before its eyes. -
-
-In other words, this principle states that the initial state \(\left (q(t_0), \dot q(t_0)\right )\) uniquely determines its evolution \(\left (q(t),\dot q(t)\right )\) for \(t > t_0\). The Picard-Lindelöf theorem6 implies that the Newtonian principle of determinacy is locally satisfied by the equations of motion of the mechanical system, i.e., second order differential equations derived from Newton’s law. -
--5 As Example 1.1 shows, the \(n\) degrees of freedom do not have to be the cartesian coordinates of the point particles. -
- --6 Also known as “Existence and uniqueness of solutions of initial value problems”, [Kna18, Theorem 3.17]. -
- - --Before taking a little detour into Lagrangian mechanics, let’s anticipate some of the upcoming concepts in a few simple cases. -
-- -Example 1.2 (Horizontal spring and pendulum). Consider an idealized system consisting of a point particle of mass \(m\) attached to a spring with stiffness \(k\), sliding on a frictionless surface. Assume that the motion is one-dimensional along the axis of the spring and let -\(x\) denote the displacement of the system from its equilibrium position, i.e., the position in which the spring is completely at rest: not compressed nor extended. We consider this system to exclude gravitational forces from the picture. -
- --Hooke’s law states that the restoring force \(F\) exerted by the spring on whatever is pulling its free end scales linearly with respect to the distance it’s been pulled: i.e., \(F(x) = - k x\), see Figure 1.3 left. According to (1.4), then, the motion of the point particle is given by +Hooke’s law states that the restoring force \(F\) exerted by an ideal spring is directly proportional to its displacement \(x\) from equilibrium. The law is expressed mathematically as \(F(x) = -kx\), where +\(k\) is the spring constant, a measure of the spring’s stiffness. The negative sign indicates that the force acts in the opposite direction of the displacement, always seeking to restore the spring to its equilibrium position. +See Figure 1.2, left panel. According to (1.4), then, the motion of the point particle is given by
\(\seteqnumber{0}{}{4}\) @@ -981,9 +898,9 @@-with the unknowns \(R\) and \(\phi \) uniquely prescribed by the initial conditions +with the unknowns \(R\in \mathbb {R}\) and \(\phi \in [0,2\pi )\) uniquely prescribed by the initial conditions
\(\seteqnumber{0}{}{6}\) @@ -1031,7 +948,7 @@-Clearly, once we know the initial conditions, the full evolution of the solution \(x(t)\) is known, in agreement with Newton’s principle of determinacy. +Once we know the initial conditions, the full evolution of the solution \(x(t)\) is known.
--Consider, now, a point particle of mass \(m\) attached to a pivot on the ceiling via a rigid rod of length \(l\). Assume the motion is frictionless and happening only in the vertical direction. Let \(x\) denote the angle of displacement of the system -from its equilibrium position, i.e., the lowest point on the arc of motion. Take the angle at equilibrium to be zero, with positive sign on the right hand side of the vertical, and negative on the left. +Consider, now, an ideal pendulum . That is, a point particle of mass \(m\) attached to a pivot on the ceiling via a rigid rod of length \(l\). Assume the motion is frictionless and happening only in a vertical plane. Let +\(x\) denote the angle of displacement of the system from its equilibrium position on this plane, i.e., the lowest point on the arc of motion. Without loss of generality, we can assume the angle at equilibrium to be zero, with positive sign on the right +hand side of the vertical, and negative on the left. See Figure 1.2, right panel.
@@ -1093,9 +982,9 @@
-Although it is possible to solve the equation of motion of the pendulum by means of elliptic integrals, this is rather cumbersome. Under certain conditions, when \(\sin x \approx x\), we can actually avoid doing this and instead we use the -equation of the spring as a model for the pendulum oscillation via the so-called called small oscillations approximation. We will come back to this later on. +Although it is possible to solve the equation of motion of the pendulum explicitl yby means of elliptic integrals [Eur24], this is rather cumbersome. In some cases, when \(\sin x \approx x\), we can simplify our life and, instead, use the equation of the spring as a model for the pendulum oscillation via +the so-called small oscillations approximation. We will come back to this later on.
- -Example 1.3 (Idealized motion of the Earth around the Sun). - Let us approximate the Sun with a point particle of mass \(M\) positioned at the origin \(\vb *{0}\in \mathbb {R}^3\) of the Euclidean space. In this solar system, with the Sun fixed at the origin, we will -describe the Earth by a point particle of mass \(m\) whose position (and motion) is described by a vector \(\vb *{x}\in \mathbb {R}^3\). + +Example 1.2 (Idealized motion of the Earth around the Sun). + Let us approximate the Sun with a point particle of mass \(M\) positioned at the origin \(\vb *{0}\in \mathbb {R}^3\) of the Euclidean space. In this model of the solar system, with the Sun fixed at the origin, +we will describe the Earth by a point particle of mass \(m\) whose position (and motion) is described by a vector \(\vb *{x}\in \mathbb {R}^3\).
-Due to our choice of coordinates, the gravitational attraction of the Sun acts in direction \(-\vb *{x}(t)\). Newton’s law of universal gravitation says that such a force is proportional to +Due to our choice of coordinates, the gravitational attraction of the Sun acts in the direction of \(-\vb *{x}(t)\). Newton’s law of universal gravitation says that such a force is proportional to
\(\seteqnumber{0}{}{8}\) @@ -1146,9 +1036,9 @@-where \(G \sim 6.674 \cdot 10^{-11} \frac {m^3}{s^2\,kg}\) is called the gravitational constant. Once we collect all the elements into Newton’s second law (1.4), we obtain the equation of motion +where \(G \sim 6.674 \cdot 10^{-11} \frac {m^3}{s^2\,kg}\) is called the gravitational constant. Newton’s second law (1.4), +then, leads to the equation of motion
\(\seteqnumber{0}{}{9}\) @@ -1172,8 +1062,8 @@This is an autonomous second order ordinary differential equation on the configuration space \(\mathbb {R}^3\setminus \big \{\vb *{0}\big \}\). Providing the initial conditions \(\vb *{x}(0)=\vb *{x}_0\) and \(\dot {\vb *{x}}(0)=\vb -*{v}_0\), we can try to explicitly solve (1.10), however in this case we could already get a lot of insight by taking a slightly different point of view. +*{v}_0\), we can try to solve (1.10), and you are welcome to try. As it is often the case, though, things can become easier by taking a step back and looking at the +problem differently.
-Behind differential equations in classical mechanics lies a surprisingly rich geometrical structure, in which conserved quantities play a special role. These can help to obtain extensive information on the solution of classical equations of motion without -the need to explicitly solve the equations (which would be, in general, impossible). +Behind differential equations in classical mechanics lies a surprisingly rich geometrical structure, in which symmetries and conserved quantities play a special role. These can help to obtain extensive information on the solution of classical equations of +motion without the need to explicitly solve the equations (which, in general, is impossible). Let’s have an initial brief look at this in the context of the Kepler problem.
-On the tangent bundle \((\mathbb {R}^3\setminus \big \{\vb *{0}\big \})\times \mathbb {R}^3\), let us define the total energy +Consider the space \(\big (\mathbb {R}^3\setminus \big \{\vb *{0}\big \}\big )\times \mathbb {R}^3\), and define there, for the moment without further justification or explanation, the total energy function
\(\seteqnumber{0}{}{10}\) @@ -1208,7 +1099,7 @@-Along the solutions of (1.10), let us define \(E(t):=E(\vb *{x}(t),\dot {\vb *{x}}(t))\) and, similarly, \(L(t)\) and \(A(t)\). Then, there exist solutions such that -\(L(t) = L(0) \neq 0\) for all times, and also \(E(t) = E(0)\) and \(A(t) = A(0)\). Furthermore, a solution that lives on the “submanifold” defined by +Along the solutions \(\vb *{x}(t)\) of (1.10), let us define \(E(t) := E(\vb *{x}(t),\dot {\vb *{x}}(t))\) and, similarly, \(L(t)\) and \(A(t)\).
++In due time we will show that, \(L(t) = L(0) \neq 0\) for all times, and also \(E(t) = E(0)\) and \(A(t) = A(0)\) are constant for all times. Furthermore, any solution that lives on the subspace defined by +
\(\seteqnumber{0}{}{13}\) + + + (x, v) ∈ R3 \ 0 × R3 | E(x, v) = E(x0 , v 0 ), L(x, v) = L(x0 , v 0 ), A(x, v) = A(x0 , v 0 ) , (1.14) + --\begin{equation} -\begin{aligned} \big \{ & (\vb *{x}, \vb *{v})\in (\mathbb {R}^3\setminus \big \{\vb *{0}\big \})\times \mathbb {R}^3 \;\mid \\ & \; E(\vb *{x},\vb *{v}) = E(\vb *{x}_0, \vb *{v}_0),\; L(\vb *{x},\vb *{v}) -= L(\vb *{x}_0, \vb *{v}_0),\; A(\vb *{x},\vb *{v}) = A(\vb *{x}_0, \vb *{v}_0) \big \}, \end {aligned} -\end{equation} + +--> -
+ + +\begin{align} +\big \{ (\vb *{x}, \vb *{v})\in \big (\mathbb {R}^3\setminus \big \{\vb *{0}\big \}\big )\times \mathbb {R}^3 \;\mid \; E(\vb *{x},\vb *{v}) = E(\vb *{x}_0, \vb *{v}_0),\; L(\vb *{x},\vb *{v}) = L(\vb *{x}_0, \vb +*{v}_0),\; A(\vb *{x},\vb *{v}) = A(\vb *{x}_0, \vb *{v}_0) \big \}, +\end{align} must be a conic of the following type
@@ -1311,7 +1205,7 @@This is an example of a central force field, and is one of the most prominent and most important examples in this course. The impatient reader can find in [Kna18, Ch. 1] a nice and -compact derivation of Kepler’s laws and the invariants above from (1.10) and its solutions. +compact derivation of Kepler’s laws and the invariants above from (1.10).
+It is a bit limiting to consider only systems with one particle and, indeed, many interesting problems in classical mechanics involve systems on interacting particles. +
-In order to leave more space to discuss Hamiltonian systems and their geometry, we will be brief7 in our account of Lagrangian mechanics and calculus of variations. For a more detailed account refer to [Arn89, Part II]. This, however, should not confuse you: Lagrangian mechanics plays a role as large as Hamiltonian mechanics in the development of classical mechanics. +In general a system of \(N\) point particles will be described by a set of \(N\) position vectors \(\vb *{x}_k = (x_k, y_k ,z_k)\) with masses \(m_k\), \(k = 1, \ldots , N\). For convenience we will denote \(\vb *{x} += (\vb *{x}_1, \ldots , \vb *{x}_N)\in \mathbb {R}^{3N}\), and \(\vb *{m} = (m_1, \ldots , m_N)\in \mathbb {R}^N\). We call \(\vb *{x}(t)\) the configuration of the system at time \(t\) in the +configuration space \(\mathbb {R}^{3N}\).
-In fact, one should not be surprised if there are many sources claiming that the most general formulation of the equations of motion in classical mechanics comes from the principle of least action or Hamilton’s variational principle. +Note that we are identifying \(\mathbb {R}^{3\times N}\), the space of \(3\times N\) matrices, and \(\mathbb {R}^{3N}\), the space of \(3N\) vectors. Namely, \(\vb *{x} = (\vb *{x}_1, \ldots , \vb *{x}_N) = (x_1, y_1, z_1, x_2, +y_2, z_2, \ldots , x_N, y_N, z_N)\) is used interchangeably as the vector of positions of the points in three-space or as the vector including all the positions of the \(N\) bodies together.
+-According to this principle, stated in 1834, the equations of motion of a mechanical system are characterized by a function \(L \equiv L(q, \dot q, t) : \mathbb {R}^n \times \mathbb {R}^n \times \mathbb {R} \to \mathbb {R}\), called -the lagrangian (function) of the system. + +Example 1.3. A system of two ideal pendulums is described by two +position vectors, so \(\vb *{x} = (\vb *{x}_1, \vb *{x}_2)\in \mathbb {R}^{6}\). Notice, however, that to describe their configuration we only need two angular variables, one for each of the pendulums. So, for all practical purposes, +the system could be completely described by \(q = (q^1, q^2) \in \mathbb {S}^1\times \mathbb {S}^1 \simeq \mathbb {T}^2\). See Figure 1.3. +
+ +
+
+
+
-7 Although not as brief as a twitter thread. +Figure 1.3: Left: a system of two decoupled pendulums in \(\mathbb {R}^3\). Right: corresponding generalized coordinates.
++The symbol \(\simeq \) usually denotes some kind of equivalence, but which specific kind is context-dependent and not often well explicited. So, let us take this opportunity to clarify the notation. What we wrote above, \(\mathbb {S}^1\times +\mathbb {S}^1 \simeq \mathbb {T}^2\), should be read as: the 2-torus, i.e., the surface of a doughnut, is diffemorphic to the product of two circles. Here, diffeomorphic is a concept from the theory of smooth +manifolds, and stands for the existence a smooth bijection between the two constructions. +
+ ++If we then think of a circle \(\mathbb {S}^1\) as the interval \([0, 2\pi )\) with the endpoints identified, the observation above is saying that the torus \(\mathbb {T}^2\) can be thought of as the square \([0, 2\pi )\times [0, 2\pi )\) with +the edges identified in the usual way and \(q = (q^1, q^2) \in [0, 2\pi )\times [0, 2\pi ) \subseteq \mathbb {R}^2\). +
+ ++Such a long story to say that, while in principle this system is 6-dimensional, for all practical purposes it is 2-dimensional. +
+ + + + + ++As exemplified by the example above, we want to set up a formalism that allows us to describe the configuration of a system in a way that is independent of the number of particles and instead captures the minimal number of parameters that are +really necessary to describe it. +
+ ++We say that a system of \(N\) particles has \(n\) degrees of freedom if we need \(n\) independent parameters to uniquely specify the system configuration. We call generalized coordinates any set of \(n\) parameters \(q = (q^1, \ldots , q^n)\) that uniquely determine the configuration of a system with \(n\) degrees of freedom, and generalized velocities their time +derivatives \(\dot q = (\dot q^1, \ldots , \dot q^n)\). The state of the system is then characterized by the set of (generalized) coordinates and velocities \((q, \dot q) = \left (q^1, \ldots , +q^n,\dot q^1, \ldots , \dot q^n\right )\). +
+ ++As Example 1.3 shows, the \(n\) degrees of freedom do not have to be the cartesian coordinates of the point particles. If you recall differential geometry, you may (correctly) guess that the +generalized coordinates will be points on some differentiable manifold \(q\in M\), their evolution will be described by a curve \(q: \mathbb {R} \to M\) parametrized by time and the state of the system will be a point in the tangent bundle \((q, +\dot q)\in TM\), i.e., \(\dot q \in T_q M\). If you don’t have any previous experience with these concepts, don’t dispare! We will take our time to introduce them as they appear throughout the course. See also section 1.4). +
+ ++We have now all the elements to translate the Newton’s principle of determinacy in mathematical terms. In 1814, Laplace [Lap51] wrote +
++ ++ ++We may regard the present state of the universe as the effect of its past and the cause of its future. An intellect which at a certain moment would know all forces that set nature in motion, and all positions of all items of which nature is composed, if +this intellect were also vast enough to submit these data to analysis, it would embrace in a single formula the movements of the greatest bodies of the universe and those of the tiniest atom; for such an intellect nothing would be uncertain and the +future just like the past would be present before its eyes. +
+
+In other words, this principle states that the initial state \(\left (q(t_0), \dot q(t_0)\right )\) uniquely determines its evolution \(\left (q(t),\dot q(t)\right )\) for \(t > t_0\). We had already observed this fact in +Example 1.1 of the pendulum, where the initial position and velocity uniquely determined the future trajectory. The Picard-Lindelöf theorem, also known as “Existence and uniqueness of solutions +of initial value problems” [Kna18, Theorem 3.17], implies that Newton’s principle of determinacy is locally satisfied by the equations of motion of the mechanical system, i.e., second order differential equations derived from Newton’s law. +
+ ++In order to leave more space to discuss Hamiltonian systems and their geometry, we will be brief in our account of Lagrangian mechanics and calculus of variations, although not as brief as this twitter thread. This, however, should not confuse you: Lagrangian mechanics plays a role as +large as Hamiltonian mechanics in the development of classical mechanics! In fact, many sources claim that the most general formulation of the equations of motion in classical mechanics comes from the principle of least action or +Hamilton’s variational principle. According to this principle, stated in 1834, the equations of motion of a mechanical system are characterized by a function +
+ \(\seteqnumber{0}{}{15}\) + + ++ +\begin{equation} +L \equiv L(q, \dot q, t) : \mathbb {R}^n \times \mathbb {R}^n \times \mathbb {R} \to \mathbb {R}, +\end{equation} + +
+ ++called the lagrangian (function) of the system. +
- -Example 1.4. The lagrangian of a non-relativistic particle with mass \(m > 0\) in a potential \(U : -\mathbb {R}^n \to \mathbb {R}\) is + +Example 1.4. The lagrangian of a mechanical system consisting of a point particle with mass \(m > 0\) +in a potential \(U : \mathbb {R}^n \to \mathbb {R}\) is
- \(\seteqnumber{0}{}{15}\) + \(\seteqnumber{0}{}{16}\) @@ -1406,7 +1424,8 @@-which is the difference between the so-called kinetic energy of the particle and its so-called potential energy. We will see where this equation comes from in Section 1.2.1. +which is the difference between the so-called kinetic energy of the particle and its so-called potential energy \(U\). We will see where this equation comes from in Section 1.2.1.
-Given a curve \(\gamma :[t_1, t_2] \to \mathbb {R}^n\), we define the action functional +Given a curve \(\gamma :[t_1, t_2] \to \mathbb {R}^n\), we define the action functional as
- \(\seteqnumber{0}{}{16}\) + \(\seteqnumber{0}{}{17}\) @@ -1445,15 +1464,33 @@+Principle of least action +
+-Assume that the configuration of our system is \(q_1 = (q_1^1, \ldots , q_1^n)\) at an initial time \(t=t_1\) and \(q_2 = (q_2^1, \ldots , q_2^n)\) at a final time \(t = t_2\). The principle of least action, or Hamilton’s -principle, states that the evolution of our system in the time interval \([t_1, t_2]\) corresponds to the curve \(q(t)\) which is the critical point of the action functional \(S[q]\) on the space of curves \(q(t)\) with \(q(t_1) = q_1\) and -\(q(t_2) = q_2\). +Consider a system with \(n\) degrees of freedom. Assume that the system is in the configuration \(q_1 = (q_1^1, \ldots , q_1^n)\) at an initial time \(t=t_1\) and \(q_2 = (q_2^1, \ldots , q_2^n)\) at a final time \(t = t_2\). +
+ ++The principle of least action, or Hamilton’s principle, states that the evolution of the system in the time interval \([t_1, t_2]\) corresponds to the curve \(q(t)\) which is the critical point of the action functional +\(S[q]\) on the space of curves \(\gamma :[t_1, t_2] \to \mathbb {R}^n\) with fixed endpoints \(\gamma (t_1) = q_1\) and \(\gamma (t_2) = q_2\).
-Let \(X\) and \(Y\) be Banach spaces8 and \(G\subset X\) an open subset of \(X\). A function \(f: G \to Y\) is called Fréchet differentiable at \(x\in G\), if there exists a bounded linear operator \(A: X \to Y\), such -that9 +To state this properly we need the concept of Banach space, that is a complete normed vector space. If \(X\) denotes a Banach space, then we will denote the corresponding norm by \(\|\cdot \|_X\). +
+ ++Let \(X\) and \(Y\) be Banach spaces and \(G\subset X\) an open subset of \(X\). A function \(f: G \to Y\) is called Fréchet differentiable at \(x\in G\), if there exists a bounded linear operator \(A: X \to +Y\), such that
\(\seteqnumber{0}{}{18}\) @@ -1473,7 +1514,7 @@-for any \(h\) in a sufficiently small neighborhood of \(0\in X\). Alternatively, you can say asymptotically as \(\|h\|_X\to 0\). -
- --As in the finite-dimensional case, \(A\) is uniquely determined and is called the Fréchet derivative of \(f\) at \(x\) and is denoted \(D f(x)\). -
--8 Banach space: complete normed vector space. +asymptotically as \(\|h\|_X\to 0\), that is, for any \(h\) in a sufficiently small neighborhood of \(0\in X\). Here, we write \(g = o(\|h\|_X)\) if \(g : X \to Y\) satisfies
--9 Here, \(g = o(\|h\|_X)\) if -
- - \(\seteqnumber{0}{}{17}\) + \(\seteqnumber{0}{}{19}\) @@ -1523,14 +1549,8 @@- +As in the finite-dimensional case, \(A\) is uniquely determined and is called the Fréchet derivative of \(f\) at \(x\) and is denoted \(D f(x)\).
- - --1. In the finite dimensional setting, Fréchet differentiability corresponds to total differentiability. As with the finite dimensional case, Fréchet’s differentiability implies the continuity of the mapping. +1. In the finite dimensional setting, Fréchet differentiability corresponds to total differentiability.
@@ -1556,8 +1576,7 @@-2. The analogy with the finite dimensional case goes further. Indeed, the chain rule, mean value theorem, implicit functions theorem, inverse mapping theorem and statements about local extremes with and -without constraints, all hold also for differentiable functions on Banach spaces. +2. As in the finite dimensional case, Fréchet differentiability implies the continuity of the mapping.
@@ -1566,7 +1585,8 @@-3. Requiring the operator \(A\) to be bounded is crucial. In \(\infty \)-dimensional normed spaces, linearity does not imply continuity. +3. The analogy with the finite dimensional case goes further. Indeed, the chain rule, mean value theorem, implicit function theorem, inverse mapping theorem and the statements about local extremes with and +without constraints, all hold also for Fréchet differentiable functions on Banach spaces.
@@ -1575,7 +1595,17 @@-4. In calculus of variation, the curve \(h\) from the small neighborhood of \(0\in X\), is usually called a variation and denoted \(\delta x\). +4. Requiring the operator \(A\) above to be bounded is crucial: in \(\infty \)-dimensional normed spaces, linearity does not imply continuity. +
+ + ++5. In calculus of variation, the curve \(h\) from the small neighborhood of \(0\in X\), is usually called a variation and denoted \(\delta x\). Don’t confuse this with a +Dirac delta function or a small real parameter.
-Let \(f: X \to \mathbb {R}\) be a (Fréchet) differentiable function and \(X_0 \subset X\) be a subspace of \(X\). Then \(\gamma _\star \) is a critical point of \(f\) with respect to \(X_0\) if +Let \(f: X \to \mathbb {R}\) be a (Fréchet) differentiable function and \(X_0 \subset X\) be a subspace of \(X\). Then \(\gamma _\star \) is a critical point of \(f\) with respect to \(X_0\) if
- \(\seteqnumber{0}{}{19}\) + \(\seteqnumber{0}{}{20}\) @@ -1627,7 +1659,7 @@\begin{equation} -X := \big \{ \gamma : [t_1, t_2] \to M \mid \gamma \mbox { is twice continuously differentiable}\big \}, +\mathcal {C}^2 := \big \{ \gamma : [t_1, t_2] \to M \mid \gamma \mbox { is twice continuously differentiable}\big \}, \end{equation}
@@ -1636,50 +1668,62 @@\begin{equation} -\|\gamma \|_X := \|\gamma \|_\infty + \|\dot \gamma \|_\infty + \|\ddot \gamma \|_\infty , +\|\gamma \|_{\mathcal {C}^2} := \|\gamma \|_\infty + \|\dot \gamma \|_\infty + \|\ddot \gamma \|_\infty , \end{equation}
-is a Banach space, and \(X_0 = \big \{h\in X \mid h(t_1) = h(t_2) = 0\big \}\) with the induced norm, is a Banach subspace of \(X\). +is a Banach space, and \(\mathcal {C}^2_0 = \big \{h\in \mathcal {C}^2 \mid h(t_1) = h(t_2) = 0\big \}\) with the induced norm, is a Banach subspace of \(\mathcal {C}^2\).
Therefore, the action defined above is a functional
- \(\seteqnumber{0}{}{22}\) + \(\seteqnumber{0}{}{23}\)\begin{equation} -S : X \to \mathbb {R},\qquad \gamma \mapsto S[\gamma ], +S : \mathcal {C}^2 \to \mathbb {R},\qquad \gamma \mapsto S[\gamma ], \end{equation}
-and the evolution of the system is described by the critical points with respect to \(X_0\) of \(S\) on the space of \(\gamma \in X\) with prescribed endpoints. +and the evolution of the system is described by the critical points with respect to \(\mathcal {C}^2_0\) of \(S\) on the space of \(\gamma \in \mathcal {C}^2\) with prescribed endpoints. +
+ + + ++1 1: Consider rewriting using q and v instead of qdot.
- + Theorem 1.1. Let \(L = L(q, \dot q, t) : \mathbb {R}^{n}\times \mathbb {R}^{n}\times \mathbb {R} \to -\mathbb {R}\) be differentiable. The equations of motion for the mechanical system with lagrangian \(L\) are given by the Euler-Lagrange equations +\mathbb {R}\) be differentiable. The equations of motion for the mechanical system with lagrangian \(L\) are given by the Euler-Lagrange equations
- \(\seteqnumber{0}{}{23}\) + \(\seteqnumber{0}{}{24}\) @@ -1724,10 +1768,61 @@+ +Remark 1.2. Be careful: when we omit the indices \(i\), we use \(\pdv {L}{q}\) to denote the gradient of +\(L\) with respect to the position variables \(q\), and \(\pdv {L}{\dot q}\) the gradient of \(L\) with respect to velocity variables \(\dot q\). In particular, \(\dot q\) in \(\pdv {L}{\dot q}\) serves as a label to denote the velocity variable, and +not as the derivative of \(q\) with respect to time. We consider \(q\) and \(\dot q\) as curves parametrised by time only when we apply \(\dv {t}\). +
+ ++Therefore, you should read the Euler-Lagrange equations above as +
+ + \(\seteqnumber{0}{}{25}\) + + + ++ + +\begin{equation} +\frac {\dd }{\dd t}\Bigg (\frac {\partial L(x,v)}{\partial v^i}\Big |_{(x,v) = (q(t),v(t))}\Bigg ) - \frac {\partial L(x,v)}{\partial x^i}\Big |_{(x,v) = (q(t),v(t))} = 0, \qquad i=1,\ldots n. +\end{equation} + + +
+ ++This explicit formulation, though precise, illustrates why such notational shortcuts are common and why I chose to adopt them. +
+ +- Proof. We need to show that any critical point of \(S\) with respect to \(X_0\) has to satisfy (1.24). + Proof. We need to show that any critical point of \(S\) with respect to \(\mathcal {C}^2_0\) has to satisfy (1.25).
-First of all observe that for small \(h\), we get +First of all observe that for small \(h\in \mathcal {C}^2_0\), we get
- \(\seteqnumber{0}{}{24}\) + \(\seteqnumber{0}{}{26}\) @@ -1850,24 +1943,24 @@-Our choice of \(h\) is arbitrary, so after a relabeling of the time, this implies10 +Our choice of \(h\) is arbitrary, so after a relabeling of the time, this implies
- \(\seteqnumber{0}{}{34}\) + \(\seteqnumber{0}{}{36}\) @@ -1875,24 +1968,29 @@-which, expanded, means +Why is that the case? Assume that (1.37) is non vanishing, then you can choose an \(h\) with vanishing endpoints but non vanishing on the support of (1.37). With such an \(h\) the integral above is strictly positive, contradicting the hypothesis.
- \(\seteqnumber{0}{}{35}\) ++Once we look at (1.37) componentwise, we obtain the Euler-Lagrange equations +
+ + \(\seteqnumber{0}{}{37}\) @@ -1900,31 +1998,19 @@-□ +as stated in the theorem, thereby concluding the proof. □
- -Remark 1.2. The solution \(q=q(t)\) of the Euler-Lagrange equations is just a critical point of the \(S\) + +Remark 1.3. The solution \(q=q(t)\) of the Euler-Lagrange equations is just a critical point of the \(S\) functional:
- \(\seteqnumber{0}{}{36}\) + \(\seteqnumber{0}{}{38}\) @@ -1964,15 +2050,15 @@- + Corollary 1.2. If the lagrangian of the system is non–degenerate, i.e., it satisfies the condition
- \(\seteqnumber{0}{}{38}\) + \(\seteqnumber{0}{}{40}\) @@ -2078,17 +2164,17 @@-Here \(\left (\Lambda ^{ij}\right ) = \left (\Lambda ^{ij}(q, \dot q, t)\right )\) are the coefficients of the inverse to the matrix (1.38), i.e., +Here \(\left (\Lambda ^{ij}\right ) = \left (\Lambda ^{ij}(q, \dot q, t)\right )\) are the coefficients of the inverse to the matrix (1.40), i.e.,
- \(\seteqnumber{0}{}{40}\) + \(\seteqnumber{0}{}{42}\) @@ -2103,7 +2189,7 @@-and it is crucial to remember that we are always using Einstein’s convention11. □ +and it is crucial to remember that we are always using Einstein’s convention2. □
-11 That is, we are summing over repeated indices, for example, \(y^{ji} x_i \equiv \sum _i y^{ji} x_i\). +2 That is, we are summing over repeated indices, for example, \(y^{ji} x_i \equiv \sum _i y^{ji} x_i\).
@@ -2134,18 +2220,18 @@- -Remark 1.3. The lagrangian of a mechanical system is defined + +Remark 1.4. The lagrangian of a mechanical system is defined only up to total derivatives. Or, in other words, the equations of motion remain unchanged if we add a total derivative to the lagrangian function:
- \(\seteqnumber{0}{}{41}\) + \(\seteqnumber{0}{}{43}\) @@ -2163,7 +2249,7 @@The action \(\widetilde S\) of a system with lagrangian \(\widetilde L\) is
- \(\seteqnumber{0}{}{42}\) + \(\seteqnumber{0}{}{44}\) @@ -2173,14 +2259,14 @@@@ -2389,7 +2475,7 @@
-We will now see how we can derive Newton’s laws from Hamilton’s principle as a consequence of the galilean principle of relativity. For an alternative discussion on this topic see [[Arn89, Chapters 1.1 and 1.2].
- + Theorem 1.3. The lagrangian of an isolated point particle in an inertial system of coordinates has the form
- \(\seteqnumber{0}{}{47}\) + \(\seteqnumber{0}{}{49}\) @@ -2436,7 +2522,7 @@- + Corollary 1.4 (Newton’s first law). In an inertial frame of reference, an isolated point particle either does not move or is in uniform motion with constant velocity.
@@ -2576,7 +2662,7 @@- Proof. It follows immediately by computing the Euler-Lagrange equations for (1.48): + Proof. It follows immediately by computing the Euler-Lagrange equations for (1.50):
- \(\seteqnumber{0}{}{52}\) + \(\seteqnumber{0}{}{54}\) @@ -2622,13 +2708,13 @@The vector \(\vb *{F}_k\) is called the force acting on the \(k\)-th point particle, and you should immediately recognize Newton’s second law in (1.56). +href="chapter-1.html#eq:newton2">1.58).
- + Exercise 1.1. Prove Newton’s third law, i.e., for each of the \(k\) point particles it holds
- \(\seteqnumber{0}{}{56}\) + \(\seteqnumber{0}{}{58}\) @@ -2884,18 +2970,18 @@- + Example 1.6. The motion of \(N\) point particles with masses \(m_1, \ldots , m_N\) in their gravitational field, is described by the lagrangian
- \(\seteqnumber{0}{}{58}\) + \(\seteqnumber{0}{}{60}\) @@ -2948,7 +3034,7 @@-whose equation of motion should remind you of (1.10) from Example 1.3. +whose equation of motion should remind you of (1.10) from Example 1.2.
-To understand better why \(T\) in (1.55) is called kinetic energy, it is useful to look at its relation with Newton’s second law (1.56): +To understand better why \(T\) in (1.57) is called kinetic energy, it is useful to look at its relation with Newton’s second law (1.58):
- \(\seteqnumber{0}{}{60}\) + \(\seteqnumber{0}{}{62}\) @@ -2987,12 +3073,12 @@-It should come as no surprise, that introducing the auxiliary variable \(y(t) = \dot x(t)\), (1.67) is equivalent to the system of first order equations +It should come as no surprise, that introducing the auxiliary variable \(y(t) = \dot x(t)\), (1.69) is equivalent to the system of first order equations
- \(\seteqnumber{0}{}{67}\) + \(\seteqnumber{0}{}{69}\) @@ -3257,16 +3343,16 @@-The solutions of (1.68) are parametric curves12 \((x(t),y(t)):\mathbb {R}\to \mathbb {R}^2\) in the \((x,y)\)-space. If \(y\neq 0\), we +The solutions of (1.70) are parametric curves3 \((x(t),y(t)):\mathbb {R}\to \mathbb {R}^2\) in the \((x,y)\)-space. If \(y\neq 0\), we can apply the chain rule, \(\frac {\dd y}{\dd t} = \frac {\dd y}{\dd x} \frac {\dd x}{\dd t}\), to get
- \(\seteqnumber{0}{}{68}\) + \(\seteqnumber{0}{}{70}\) @@ -3280,15 +3366,15 @@-Reasoning formally13 for a moment, we can rephrase this as +Reasoning formally4 for a moment, we can rephrase this as
- \(\seteqnumber{0}{}{69}\) + \(\seteqnumber{0}{}{71}\) @@ -3301,7 +3387,7 @@-Getting rid of time and considering equation (1.69) comes with a price, the solution is now an implicit curve \(y(x)\), but also with a huge advantage: this new equation can +Getting rid of time and considering equation (1.71) comes with a price, the solution is now an implicit curve \(y(x)\), but also with a huge advantage: this new equation can be solved exactly!
@@ -3309,13 +3395,13 @@- + Theorem 1.7. If the lagrangian of the mechanical system does not explicitly depend on time, \(L = L(q, \dot q)\), then the energy of the system
- \(\seteqnumber{0}{}{77}\) + \(\seteqnumber{0}{}{79}\) @@ -3656,7 +3742,7 @@- + Example 1.9. Let’s consider \(N\) point particles in physical space with -natural lagrangian \(L = T - U\) as in (1.55). Then, +natural lagrangian \(L = T - U\) as in (1.57). Then,
- \(\seteqnumber{0}{}{80}\) + \(\seteqnumber{0}{}{82}\) @@ -3765,13 +3851,13 @@- -Remark 1.7. This is probably a good point to discuss the question: why does nature want to minimize the -action? And why the lagrangian is of the form (1.55)? + +Remark 1.8. This is probably a good point to discuss the question: why does nature want to minimize the +action? And why the lagrangian is of the form (1.57)?
-Theorem 1.7 tells us that the total energy is conserved, and (1.83) tells us that for closed systems +Theorem 1.7 tells us that the total energy is conserved, and (1.85) tells us that for closed systems this implies that the energy is transferred back and forth between the kinetic and the potential components. We saw towards the end of Section 1.2.1 that while the kinetic energy measures how much the system is moving around, the potential energy measures the capacity of the system to change: its name can be intended as ‘potential’ in the sense of yet unexpressed possibilities.
@@ -3883,8 +3969,8 @@-To begin with, we can use (1.86) to integrate the equations of motion +To begin with, we can use (1.88) to integrate the equations of motion
- \(\seteqnumber{0}{}{86}\) + \(\seteqnumber{0}{}{88}\) @@ -3971,7 +4057,7 @@- -Remark 1.8. Integral curves carry a lot more information than what immediately meets the eye. Observe + +Remark 1.9. Integral curves carry a lot more information than what immediately meets the eye. Observe for example that there is a remarkable relation between the period of an oscillation and the area of the integral curve \(H(x,y) = E\): if \(A\) is the area enclosed by the integral curve, then the period \(\mathcal {T}\) can be obtained as
- \(\seteqnumber{0}{}{91}\) + \(\seteqnumber{0}{}{93}\) @@ -4123,12 +4209,12 @@- + Example 1.10. Looking back at Example 1.2, the lagrangian of a pendulum of length \(l\) and mass \(m\) is given by +href="chapter-1.html#ex:sprPen">1.1, the lagrangian of a pendulum of length \(l\) and mass \(m\) is given by
- \(\seteqnumber{0}{}{94}\) + \(\seteqnumber{0}{}{96}\) @@ -4214,16 +4300,16 @@thus \(E = \frac {ml^2 \dot x^2}2 - mgl \cos x\). As one can see in Figure 1.5, the motion is bounded for \(|E| \leq mgl\). For any such motion, the angle of maximal oscillation -corresponds to a solution of \(U(x) = E\), thus is given by \(E = -mgl \cos x^*_E\) for some \(x^*_E\). Finally, (1.90) implies that the period of oscillation is +corresponds to a solution of \(U(x) = E\), thus is given by \(E = -mgl \cos x^*_E\) for some \(x^*_E\). Finally, (1.92) implies that the period of oscillation is
- \(\seteqnumber{0}{}{95}\) + \(\seteqnumber{0}{}{97}\) @@ -4303,17 +4389,17 @@-Equation (1.98) implies that for small oscillations, the period of oscillation is also independent of the amplitude (and thus on the energy \(E\)). +Equation (1.100) implies that for small oscillations, the period of oscillation is also independent of the amplitude (and thus on the energy \(E\)). Which is the same result that you get by approximating the potential for small \(x\) by a second order polynomial
- \(\seteqnumber{0}{}{98}\) + \(\seteqnumber{0}{}{100}\) @@ -4336,13 +4422,13 @@-which we already solved in Example 1.2. In this specific case \(\omega = \sqrt {g/l}\) and the corresponding period of oscillation is +which we already solved in Example 1.1. In this specific case \(\omega = \sqrt {g/l}\) and the corresponding period of oscillation is
- \(\seteqnumber{0}{}{100}\) + \(\seteqnumber{0}{}{102}\) @@ -4392,7 +4478,7 @@@@ -4430,7 +4516,7 @@
-
+
Example 1.11 (Double pendulum). A double pendulum consists of two point particles of masses \(m_1\) and \(m_2\), connected by massless rods of lengths \(l_1\) and \(l_2\). The first pendulum is attached to the ceiling by a fixed pivot while the second
one is attached to the point particle of the first pendulum.
@@ -4445,13 +4531,13 @@ 1.3.3 Fun with the
The lagrangian \(L_1 = T_1 - U_1\) of the first particle is the same as for a simple pendulum:
The full lagrangian is then given by
- \(\seteqnumber{0}{}{105}\) + \(\seteqnumber{0}{}{107}\) @@ -4557,11 +4643,11 @@- + Example 1.12 (Lissajous Figures). Consider the system of independent harmonic oscillators
- \(\seteqnumber{0}{}{108}\) + \(\seteqnumber{0}{}{110}\) @@ -4640,13 +4726,13 @@@@ -4699,13 +4785,13 @@
-This is contained in a rectangle \([-R_1,R_1]\times [-R_2,R_2]\). When \(\omega \) is irrational, the trajectory fills the rectangle, whereas they are closed curves inside the rectangle when it is rational. See also [[Kna18, Exercise 6.34] and [Arn89, Chapter 2.5, Example 2].
@@ -4737,7 +4823,7 @@- + Exercise 1.3 (Huygens problem). Determine the form of even potentials, \(U(x) = U(-x)\), giving rise to isochronal oscillations, i.e., whose period \(\mathcal {T}\) does not depend on \(E\). (Beware, this is hard!)
@@ -4746,13 +4832,13 @@
-
+
Exercise 1.4 (The isoperimetric problem). In the
\((x,y)\)-plane in \(P\subset \mathbb {R}^3\) you are given a curve \(\gamma :[t_1,t_2]\to P\) connecting the points \((x_1, y_1, 0)\) and \((x_2, y_2, 0)\). Revolve this curve around the \(x\)-axis. For which curve does the
corresponding surface of revolution have minimal area?
@@ -4804,7 +4890,7 @@
-
+
Exercise 1.5 (The brachistochrone problem). Let \(P\),
\(Q\) be two given points in the vertical \((x,y)\)-plane, where \(Q\) lies beneath \(P\). A particle, subject to constant gravitational acceleration pointing downwards, moves from \(P\) to \(Q\), starting at rest. Determine the curve along which this
motion takes the shortest time.
@@ -4818,8 +4904,8 @@ 1.3.3 Fun with the
1.3.3 Fun with the
-
1.4 Euler-Lagrange equations on smooth manifolds
-
+1.4 Euler-Lagrange equations on smooth manifolds
+
@@ -4833,12 +4919,12 @@ 1.4 Euler-Lagrange
\(\mathbb {R}^{3N}\). Assume that we already know that the system has \(n\) degrees of freedom. If we describe the motion in terms of generalized coordinates
1.4 Euler-Lagrange
a direct application of the chain rule will allow us to rewrite the lagrangian in terms of these new coordinates as
-The strange looking object \(\dd s^2\), called line element or first fundamental form, is just a short-hand notation for the \(g\)-dependent scalar product of generalized velocities that we obtained above14: \(\dd s^2 = g_{kl}(q)\, -\dd q^k \dd q^l\). For example, in cartesian coordinates, we have +The strange looking object \(\dd s^2\), called line element or first fundamental form, is just a short-hand notation for the \(g\)-dependent scalar product of generalized velocities that we obtained above5: \(\dd s^2 = g_{kl}(q)\, \dd +q^k \dd q^l\). For example, in cartesian coordinates, we have
- \(\seteqnumber{0}{}{118}\) + \(\seteqnumber{0}{}{120}\) @@ -4996,12 +5082,12 @@Let’s consider the more general situation of local coordinates \((q^1, \ldots , q^n)\) on a smooth manifold \(M\). We will only recall the main terminology here, for a review of these concepts refer to [Arn89, Chapter 4.18], [Kna18, Appendix A], [MR99, -Chapter 4], your favourite book of differential geometry15 or my beautiful16 lecture notes [Ser20]. +Chapter 4], your favourite book of differential geometry6 or my beautiful7 lecture notes [Ser20].
-Local coordinates, or charts, on \(M\) are local homeomorphisms17 +Local coordinates, or charts, on \(M\) are local homeomorphisms8
- \(\seteqnumber{0}{}{121}\) + \(\seteqnumber{0}{}{123}\) @@ -5112,7 +5198,7 @@-is the decomposition of the vector \(v\) in the local basis induced by the coordinate chart. This in particular means that \(\frac {\partial }{\partial q^i}\) is just a notation for the \(i\)-th basis versor20 in \(T_q M\). Mapping +is the decomposition of the vector \(v\) in the local basis induced by the coordinate chart. This in particular means that \(\frac {\partial }{\partial q^i}\) is just a notation for the \(i\)-th basis versor11 in \(T_q M\). Mapping \(\frac {\partial }{\partial q^i}\) to the \(i\)-th Euclidean basis versor \(e_i\) of \(\mathbb {R}^n\) allows you to consider tangent vectors in coordinates as genuine euclidean vectors applied at a specific point \(q\).
@@ -5127,12 +5213,12 @@-which have the exact same form as (1.24). Under the time-independence and non–degeneracy conditions, the lagrangian defines a dynamical system on the tangent +which have the exact same form as (1.25). Under the time-independence and non–degeneracy conditions, the lagrangian defines a dynamical system on the tangent bundle \(TM\).
@@ -5402,12 +5488,12 @@-The length of curves on \(M\) is then defined as in (1.118) and angles between tangent vectors are defined using a generalization of the law of cosines where the Euclidean +The length of curves on \(M\) is then defined as in (1.120) and angles between tangent vectors are defined using a generalization of the law of cosines where the Euclidean scalar product is replaced by the inner product on \(T_qM\):
- \(\seteqnumber{0}{}{133}\) + \(\seteqnumber{0}{}{135}\) @@ -5445,12 +5531,12 @@-21 Truth be told, we are hiding some extra complications involving infinite dimensional manifolds here, for a full account of the theory involved you can refer to [12 Truth be told, we are hiding some extra complications involving infinite dimensional manifolds here, for a full account of the theory involved you can refer to [MR99, Chapters 7 and 8].
@@ -5508,13 +5594,13 @@-Solutions of (1.138) are the geodesic curves on the riemannian manifold \(M\). Locally, geodesics are not just minimizing the action \(S\), but also the lengths! +Solutions of (1.140) are the geodesic curves on the riemannian manifold \(M\). Locally, geodesics are not just minimizing the action \(S\), but also the lengths!
-1. Derive formula (1.83) for a natural lagrangian on a riemannian manifold \((M,g)\). +1. Derive formula (1.85) for a natural lagrangian on a riemannian manifold \((M,g)\).
@@ -5617,7 +5703,7 @@
-2. Given any lagrangian of the form \(L = L(q, \dot q)\) on \(TM\), show that the definition of energy (1.78) does not depend on
+2. Given any lagrangian of the form \(L = L(q, \dot q)\) on \(TM\), show that the definition of energy (1.80) does not depend on
the choice of local coordinates on \(M\).
Hint: look at Exercise 1.6.
3. Consider the following free lagrangian on a riemannian manifold \((M, g)\):
- \(\seteqnumber{0}{}{139}\) + \(\seteqnumber{0}{}{141}\)@@ -5666,23 +5752,23 @@
- + Example 1.13. Let us consider a spherical pendulum of mass \(m\) and length \(l\), i.e., a point particle of mass \(m\) moving without friction on the surface of a sphere of radius \(l\). Let’s ignore gravity for the moment.
The system has two degrees of freedom, its configuration space \(M\) is a 2-sphere \(S^2\) of radius \(l\). Using spherical coordinates \((\theta , \phi )\) – see also Figure 1.8 – and (1.121), we find the free lagrangian +class="textup">(1.123), we find the free lagrangian
- \(\seteqnumber{0}{}{140}\) + \(\seteqnumber{0}{}{142}\) @@ -5714,9 +5800,9 @@- + Exercise 1.9. Start from a free lagrangian in cartesian coordinates. As we are imposing that the particle -lives on the surface of the sphere, change variables using the appropriate spherical coordinates to derive (1.141). +lives on the surface of the sphere, change variables using the appropriate spherical coordinates to derive (1.143).
@@ -5765,15 +5851,15 @@
-In analogy with our discussion of natural lagrangians (1.55), on riemannian manifolds we call natural the lagrangians of the form +In analogy with our discussion of natural lagrangians (1.57), on riemannian manifolds we call natural the lagrangians of the form
- \(\seteqnumber{0}{}{141}\) + \(\seteqnumber{0}{}{143}\) @@ -5796,10 +5882,10 @@- -Remark 1.9. It is often necessary to deal with mechanical systems in which the interactions between + +Remark 1.10. It is often necessary to deal with mechanical systems in which the interactions between different bodies take the form of constraints. These constraints can be very complicated and hard to incorporate when working directly with the differential equations of motion. It is often easier to construct a lagrangian out of the kinetic and -potential energy of the system and derive the correct equations of motion from there. Mechanical lagrangians will often be of the form (1.142). +potential energy of the system and derive the correct equations of motion from there. Mechanical lagrangians will often be of the form (1.144).
- + 1.2 Hamilton’s variational principle
- + 1.3 First steps with conserved quantities
- + 1.4 Euler-Lagrange equations on smooth manifolds
- + 2.2 The spherical pendulum
- + 2.3 Intermezzo: small oscillations
- + 2.4 Motion in a central potential
- + 2.5 D’Alembert principle and systems with constraints
- + 3.1 The Legendre Transform in the euclidean plane
- + 3.2 Intermezzo: cotangent bundle and differential forms
- + 3.3 The Legendre Transform
- + 3.4 Poisson brackets and first integrals of motion
- + 3.5 Variational principles of hamiltonian mechanics
- + 3.6 Canonical transformations
- + 3.7 The symplectic structure on the cotangent bundle
- + 4.1 Lagrangian submanifolds
- + 4.2 Canonical transformations revisited
- + 4.3 Hamilton-Jacobi equations
- + 4.4 The Liouville-Arnold theorem
- + 5 Hamiltonian perturbation theory
- + 5.1 Small oscillations revisited
- + 5.3 A brief look at KAM theory
++ +Index +
+ +@@ -676,7 +682,7 @@
- + Example 2.1 (kinetic momentum). Consider a mechanical system with \(n\) degrees of freedom described by the lagrangian \(L=L(q,\dot q)\), \(q=(q^1,\ldots ,q^n)\). If there is \(i\) such that \(L\) does not depend on \(q^i\), i.e.
@@ -894,7 +900,7 @@1 The operation \(\Phi ^*:C^\infty (TM) \to C^\infty (TM)\) defined here is a special case of a more general operation, called pullback, which we will see again in different settings in future chapters. @@ -987,7 +993,7 @@
- + Remark 2.1. To any one–parameter group of diffeomorphism we can associate a smooth vector field \(X : q\in M \mapsto X(q)\in T_qM\) by
@@ -1008,7 +1014,7 @@- + Exercise 2.1. Show that \(\Phi _s\) as defined in (2.11) is a one–parameter group of diffeomorphisms on \(M\) for \(s,t \in (-\epsilon ,\epsilon )\) such that \(|s + t| < \epsilon \).
@@ -1123,7 +1129,7 @@
3 For a pictorial glimpse to her life, look for #Noethember on instagram or 2.1 Noether theore
\begin{equation}
-I(q,\dot q) := p_i X^i, \quad p_i(q, \dot q) := \frac {\partial L}{\partial \dot q^i}, \quad X(q) := \frac {\dd }{\dd s} \Phi _s(q) \Big |_{s=0}.
+I(q,\dot q) := p_i X^i, \quad p_i(q, \dot q) := \frac {\partial L}{\partial \dot q^i}, \quad X(q) := \dv {s} \Phi _s(q) \Big |_{s=0}.
\end{equation}
@@ -1186,7 +1192,7 @@ 2.1 Noether theore
- Proof. By definition, see (1.76), \(I\) is a first integral if it is constant on the + Proof. By definition, see (1.78), \(I\) is a first integral if it is constant on the phase curves. Let’s compute its time derivative,
\(\seteqnumber{0}{}{12}\) @@ -1304,7 +1310,7 @@- + Example 2.2 (kinetic momentum - reprise). Let’s revisit Example 2.1. For a cyclic coordinate \(q^i\), the lagrangian is invariant with respect to the one–parameter group of translations along the \(q^i\) direction:
@@ -1370,7 +1376,7 @@
-
+
Remark 2.2. Noether theorem is a more general result related to symmetries in classical field theories than
what we presented here. The proof of the more general Noether theorem is obtained by reducing the system to a system with one cyclic variable. This can be done by locally rectifying4 the vector field \(X\) on \(M\) and observing that,
by Exercise 1.6, the sum \(p_i X^i\) does not depend on the choice of coordinates on \(M\).
@@ -1381,7 +1387,7 @@ 2.1 Noether theore
4 See [Ser20, Chapter 3.5]. @@ -1400,7 +1406,7 @@
-
+
Exercise 2.2. Assume that the one–parameter group of diffeomorphisms \(\Phi _s\) is not a
symmetry for \((M,L)\) in the sense of our definition, but preserves the action \(S\) associated to the lagrangian. In that case, the invariance condition (2.17) holds only up to total derivatives [GF00, Chapter 4.20], i.e., there is a function \(f(q)\) on \(M\) such that
@@ -1471,7 +1477,7 @@
-
+
Remark 2.3. There are also discrete symmetries in nature which don’t depend on a continuous parameter.
The so-called parity, for example, is the invariance under reflection, \(\vb *{x} \mapsto -\vb *{x}\). These types of symmetries give rise to conservation laws in quantum physics but not in classical physics.
-Consider a system of \(N\) particles with lagrangian (1.55),
+Consider a system of \(N\) particles with lagrangian (1.57),
2.1 Noether theore
2.1 Noether theore
-
2.1.1 Homogeneity of space: total momentum
-
+2.1.1 Homogeneity of space: total momentum
+
2.1.1 Homogeneity
5 It may seem here that we forgot to push forward \(\dot x\) by \(\Phi _s\), however the pushforward \((\Phi _s)_* = \Id \) by \(\Phi _s\) acts on \(\dot x\) as the identity. @@ -1802,7 +1808,7 @@
6 Beware, here (and a few other times below) \(M\) denotes a constant, to not be confused with the base manifold. What is what should be inferrable from the constants. @@ -1821,7 +1827,7 @@
- + Exercise 2.3. Consider, as in Example 1.7 and Exercise 1.2, a closed system of charged point particles in the presence of a uniform constant magnetic field \(\vb *{B}\). Define
@@ -1860,8 +1866,8 @@- + Proposition 2.3. Let \(A(s)\) be a one–parameter family of orthogonal matrices of the form (2.34). Then,
@@ -1997,7 +2003,7 @@- + Exercise 2.4 (Euler theorem). Given an antisymmetric matrix
@@ -2216,7 +2222,7 @@- + Theorem 2.4. If a closed system of \(N\) particles is invariant with respect to rotations, then the components of its total angular momentum
@@ -2253,7 +2259,7 @@- + Example 2.4. Consider a natural lagrangian (1.55) with an axial symmetry, i.e., whose potential is invariant by simultaneous rotations around a fixed axis. For convenience let’s say that such axis corresponds to the \(z\) axis. +href="chapter-1.html#eq:mechlag">1.57) with an axial symmetry, i.e., whose potential is invariant by simultaneous rotations around a fixed axis. For convenience let’s say that such axis corresponds to the \(z\) axis. Looking back at the proof above, we can infer that the conserved quantity will not be the full total momentum \(\vb *{M} = (M_x, M_y, M_z)\) but instead only its \(z\) component:
@@ -2421,8 +2427,8 @@@@ -2441,12 +2447,12 @@
-We have mentioned in Remark 1.3 that the two lagrangians \(L\) and \(\alpha L\), \(\alpha \neq 0\), give rise to the same equations of motion. This does not look like a symmetry in +We have mentioned in Remark 1.4 that the two lagrangians \(L\) and \(\alpha L\), \(\alpha \neq 0\), give rise to the same equations of motion. This does not look like a symmetry in our sense, but it is worth investigating its meaning in light of what we have seen so far.
- + Theorem 2.5. Let \(L\) be a natural lagrangian \(L=T-U\) as in (1.55) whose potential is a homogeneous function of degree \(k\): +href="chapter-1.html#eq:mechlag">1.57) whose potential is a homogeneous function of degree \(k\):
\(\seteqnumber{0}{}{54}\) @@ -2518,7 +2524,7 @@- + Remark 2.4. The reasoning above can be repeated for any Lagrangian \(L = T - U\) where \(T\) is homogeneous of degree \(k\) and \(U\) is homogeneous of degree \(l\), obtaining a similar result.
@@ -2658,7 +2664,7 @@- + Example 2.5 (Harmonic oscillator). The potential of the harmonic oscillator
@@ -2701,9 +2707,9 @@- + Example 2.6 (Kepler’s third law). Recall Kepler’s problem -lagrangian from Examples 1.3 and 1.6: +lagrangian from Examples 1.2 and 1.6:
\(\seteqnumber{0}{}{61}\) @@ -2763,8 +2769,8 @@- + Remark 2.5. Be careful here: Euler-Lagrange equations are derived under the assumption that the coordinates are independent. If you replace the quantity \(I\) in the lagrangian itself, you will get an incorrect result!
@@ -2985,7 +2991,7 @@- + Exercise 2.5. Use the phase portrait to describe the behavior of \(\theta (t)\), and the motion of the pendulum. Make sure to use the correct intervals of definition of \(\phi \) and \(\theta \).
@@ -3006,8 +3012,8 @@Let’s revisit what we saw in Sections 1.3.1 and 1.3.3 in slightly more generality and in a more organic way. Let’s consider the Newton equation for a -closed system with one degree of freedom in the presence of conservative forces, i.e., an equation like (1.67): +closed system with one degree of freedom in the presence of conservative forces, i.e., an equation like (1.69):
\(\seteqnumber{0}{}{71}\) @@ -3318,7 +3324,7 @@- + Theorem 2.6. A system performing small oscillations is the direct product of \(n\) one-dimensional systems performing small oscillations.
@@ -3381,7 +3387,7 @@- + Theorem 2.7. The system (2.78) has \(n\) characteristic oscillations, the directions of which are pairwise orthogonal with respect to the scalar product given by the kinetic energy.
@@ -3403,7 +3409,7 @@- + Corollary 2.8. Every small oscillation is a sum of characteristic oscillations.
@@ -3424,7 +3430,7 @@- + Remark 2.6. Note that, in general, it is not possible to simultaneously diagonalize three symmetric matrices.
@@ -3436,8 +3442,8 @@- + Remark 2.7. As opposed to general systems of linear differential equations, resonance terms of the form \(t^\alpha e^{i \omega t}\), \(\alpha \in \big \{1,2,\ldots ,n\big \}\), do not arise in lagrangian systems (even in the case of eigenvalues with multiplicity).
@@ -3547,8 +3553,8 @@@@ -3695,8 +3701,8 @@
@@ -3941,7 +3947,7 @@
- + Remark 2.8. We have already seen an example of this in Section 2.3.3: the normal mode \(\vb *{\xi }_1=(1,1,1)^T\) of the triatomic molecule is the zero mode corresponding to the invariance of the system with respect to horizontal translations.
@@ -3959,8 +3965,8 @@- + Example 2.7 (The two-body problem). Let’s go back to Examples 1.3 and 1.6. Consider a closed systems of two point particles with masses \(m_{1,2}\). We now +id="ex:kepler2"> Let’s go back to Examples 1.2 and 1.6. Consider a closed systems of two point particles with masses \(m_{1,2}\). We now know that, in an inertial frame of reference, their natural lagrangian must have the form
@@ -4185,7 +4191,7 @@- + Theorem 2.9. The trajectories of (2.107) are planar.
@@ -4197,7 +4203,7 @@@@ -4583,7 +4589,7 @@
-Figure 2.1: The trajectories are confined in an annular region [LL76] +Figure 2.1: The trajectories are confined in an annular region [LL76]
- + Exercise 2.7. Consider the integral (2.119), with \(r_0 = r_m\) and \(r = r_M\), one can use it to define the quantity
@@ -4668,12 +4674,12 @@-Let’s go back to Example 2.7, and replace \(U\) with the Newtonian gravitational potential \(k/\|\vb *{x}\|\) (see Examples 1.3 and 2.7, and replace \(U\) with the Newtonian gravitational potential \(k/\|\vb *{x}\|\) (see Examples 1.2 and 1.6). The effective potential (2.120), in this case is
\(\seteqnumber{0}{}{123}\) @@ -4727,7 +4733,7 @@@@ -4842,7 +4848,7 @@
-We can now confirm the last claim in Example 1.3. Indeed, with some knowledge about conic sections one can conclude that (1.2. Indeed, with some knowledge about conic sections one can conclude that (2.129) is an ellipse for \(E<0\), when \(0<e<1\). The semiaxes of the ellipse are then given by
@@ -4868,7 +4874,7 @@@@ -5145,7 +5151,7 @@
- + Exercise 2.8. Obtain the following parametrization for the hyperbolic motion for \(E>0\).
@@ -5202,7 +5208,7 @@- + Exercise 2.9. Obtain the following parametrization for the parabolic motion for \(E=0\).
@@ -5265,7 +5271,7 @@- + Exercise 2.10. Let \(0<e<1\) and \(\tau = \theta - e\sin \theta \). Show that \(\sin \theta \) and \(\cos \theta \) can be expanded in Fourier series as
@@ -5363,7 +5369,7 @@-In Example 1.3, we mentioned a third conserved quantity for the Kepler problem. +In Example 1.2, we mentioned a third conserved quantity for the Kepler problem.
- + Exercise 2.11. Show that the components of the Laplace-Runge-Lenz vector
@@ -5415,7 +5421,7 @@- + Exercise 2.12. Consider a small perturbation of the gravitational potential \(U(r) = -\frac kr\) of the form
@@ -5453,8 +5459,8 @@-Let’s try again. Consider now a natural lagrangian (1.55) +Let’s try again. Consider now a natural lagrangian (1.57)
\(\seteqnumber{0}{}{150}\) @@ -5528,7 +5534,7 @@- + Remark 2.9. We call the constraints holonomic if the functions that define them do not depend on the velocities (they can depend on time, their treatment is analogous to what we will do in this section). More general constraints, also taking into account the velocities, are called non holonomic and will not be treated in this course.
@@ -5577,9 +5583,9 @@- + Theorem 2.11. The equations of motion of a mechanical system of \(N\) particles with natural lagrangian (1.55) under the effect of \(l\) independent holonomic constraints (2.152) have the form +class="textup">(1.57) under the effect of \(l\) independent holonomic constraints (2.152) have the form
\(\seteqnumber{0}{}{152}\) @@ -5614,7 +5620,7 @@
7 For a brief introduction on their meaning, I recommend this twitter thread or this 2.5 D’Alembert
-
+
Remark 2.10. The theorem above is often stated as follows: in a constrained system, the total work of the
constraint forces on any virtual variations, i.e., vectors \(\delta \vb *{x}\) tangent to the submanifold \(Q\), is zero. One can understand this by observing that for such tangent vectors
-
+
Theorem 2.12. The evolution of of a mechanical system on \(N\) particles with natural lagrangian (1.55) under the effect of \(l\) independent holonomic constraints (2.152) can be described by
+class="textup">(1.57) under the effect of \(l\) independent holonomic constraints (2.152) can be described by
mechanical system on \(TQ\) with lagrangian
-
+
Exercise 2.13. Show that the equations of motion of the constrained motion on \(Q\) are given by the
Euler-Lagrange equations of the lagrangian (2.162).
-
+
Example 2.8. Every submanifold \(Q\subset \mathbb {R}^m\) of the euclidean space is regular with
respect to the free lagrangian
-
+
Exercise 2.14. Consider the lagrangian \(L=\frac 12 g_{ij}(q)\dot q^i \dot q^j\) on the tangent
bundle of a riemannian manifold \(Q\) with a metric \(\dd s^2 = g_{ij}(q)\dd q^i \dd q^j\). Show that the solutions \(q=q(t)\) of the Euler-Lagrange equations for \(L\) satisfy \(\langle \dot q, \dot q\rangle = \mathrm {const}\).
-
+
Exercise 2.15 (Clairaut’s relation). Given a function
\(f(z): [a,b] \to \mathbb {R}_+\), consider the surface of revolution in \(\mathbb {R}^3\) implicitly defined by
-
+
Example 2.9 (Double pendulum revisited). Let’s see if we
can re-derive the double pendulum lagrangian using D’Alembert principle. We have a system of two point masses with masses \(m_1\) and \(m_2\) connected by massless rigid rods: one of length \(l_2\) connecting them to each other and one of
length \(l_1\) connecting the first to a pivot on the ceiling. We assume that frictions are absent and that the double pendulum is constrained to move in a single plane.
@@ -6168,7 +6174,7 @@
-
+
Exercise 2.16. Use D’Alembert principle to compute the constraint force acting on the spherical
pendulum9 under the effect of the gravitational potential.
9 Section 2.2.
diff --git a/chapter-3.html b/chapter-3.html
index dbcc843..5b20449 100644
--- a/chapter-3.html
+++ b/chapter-3.html
@@ -234,7 +234,7 @@
-
+
1.2 Hamilton’s variational principle
-
+
1.3 First steps with conserved quantities
-
+
1.4 Euler-Lagrange equations on smooth manifolds
-
+
2.2 The spherical pendulum
-
+
2.3 Intermezzo: small oscillations
-
+
2.4 Motion in a central potential
-
+
2.5 D’Alembert principle and systems with constraints
-
+
3.1 The Legendre Transform in the euclidean plane
-
+
3.2 Intermezzo: cotangent bundle and differential forms
-
+
3.3 The Legendre Transform
-
+
3.4 Poisson brackets and first integrals of motion
-
+
3.5 Variational principles of hamiltonian mechanics
-
+
3.6 Canonical transformations
-
+
3.7 The symplectic structure on the cotangent bundle
-
+
4.1 Lagrangian submanifolds
-
+
4.2 Canonical transformations revisited
-
+
4.3 Hamilton-Jacobi equations
-
+
4.4 The Liouville-Arnold theorem
-
+
5 Hamiltonian perturbation theory
-
+
5.1 Small oscillations revisited
-
+
5.3 A brief look at KAM theory
+
+Index
+
@@ -717,8 +723,8 @@
@@ -917,7 +923,7 @@
1 Metaphorically, but I should probably add a sketch here.
@@ -1212,8 +1218,8 @@
@@ -1277,7 +1283,7 @@
-
+
Theorem 3.1. The Legendre transform of a non–degenerate lagrangian is a local diffeomorphism.
-We call hamiltonian of a mechanical system with lagrangian \(L\), the Legendre transform \(H(q,p)\) of its energy \(E(q,\dot q)\) from (1.78). In local
+We call hamiltonian of a mechanical system with lagrangian \(L\), the Legendre transform \(H(q,p)\) of its energy \(E(q,\dot q)\) from (1.80). In local
coordinates the hamiltonian of the system is given by
-
+
Remark 3.1. This is what is meant when you read that the hamiltonian is the lift of the total energy \(E(q,
\dot q)\) of the system to a function \(H(q,p)\) on the cotangent bundle \(T^*M\).
-
+
Remark 3.2. This is where the literature can disagree. Some textbooks define \(H\) to be the Legendre
transform of \(L\) instead. In this way the definition can be rewritten3 as
3 This is very nicely discussed in this blog post.
@@ -1515,7 +1521,7 @@
-
+
Remark 3.3. Perhaps not surprisingly, the Legendre transform also comes with some interesting connections
to control theory and differential geometry. Here I will focus on the latter. For a more detailed discussion about the relations to control theory, you can refer to [GF00, Appendix II].
We will not go into the details here and instead only point out the bare-bones of the connection. Optimization problems can often be reduced to find a critical point of some functional, like the action, with respect to some extra external constraints,
@@ -1574,7 +1580,7 @@
-
+
Theorem 3.2 (Hamilton’s equations).
Euler-Lagrange equations for a lagrangian \(L(q,\dot q)\) are equivalent to Hamilton’s equations
-
+
Example 3.1 (A particle in a potential). Let’s start with a simple prototypical example: a particle in \(\mathbb {R}^3\) moving under the influence of a potential \(U(\vb *{x})\). We are, by now, very accustomed to the corresponding natural lagrangian
-
+
Example 3.2 (The hamiltonian for a natural lagrangian on a riemannian
manifold). Let \(L\) be a free lagrangian
-
+
Exercise 3.1. Show that a simple substitution allows to recover Euler-Lagrange equations of
Exercise 1.7, which gives the geodesic flow on the tangent bundle \(TM\).
-
+
Remark 3.4. The previous example provides a good opportunity to discuss the
nature of the momentum from yet another perspective. Let \((M, g)\) be a riemannian manifold. The riemannian metric, or riemannian metric tensor, is a bilinear form \(\langle \cdot , \cdot \rangle _g\) on the tangent spaces of \(M\). In
local coordinates \(q\in M\), we can associate to the metric a matrix \(g_{ij}(q)\). With an abuse of notation, we say that \(g_{ij}(q)\) is a \((0,2)\)-tensor.
@@ -2213,7 +2219,7 @@
4 The isomorphisms between the tangent and cotangent spaces defined here are commonly known as musical isomorphisms: the flat and sharp maps respectively correspond to the mapping \(\beta
@@ -2234,7 +2240,7 @@
-
+
Example 3.3. We saw in Example 3.1 that the
hamiltonian of a free particle of mass \(m\) is, in cartesian coordinates,
-
+
Exercise 3.2 (Inverse Legendre transform). Show that the
hamiltonian \(H\) defined by (3.26) satisfies the non–degeneracy condition
-
+
Exercise 3.3. Show that the Legendre transform of a non–degenerate
time-dependent lagrangian \(L(q,\dot q, t)\) is well defined, the corresponding hamiltonian is
-
+
Example 3.4 (A particle in an electromagnetic field). We saw the Lagrangian for charged particles in an electromagnetic field in Example 1.7 and in Exercise 1.2. In
the simpler case of a single particle, the lagrangian takes the form
@@ -2674,7 +2680,7 @@
-
+
Example 3.5 (A particle in a constant magnetic field). An
important special case of Example 3.4 is the case of a uniform magnetic field pointing in the \(z\)-direction: \(\vb * B = (0,0,B)\). One convenient choice of vector potential for \(\vb * B\) is
-
+
Example 3.6 (The relativistic hamiltonian). If \(c>0\)
denotes the speed of light, the hamiltonian \(H:\mathbb {R}^3\times \mathbb {R}^3\to \mathbb {R}\) of a free relativistic particle of mass \(m>0\) is
-
+
Exercise 3.4. Show that the definition of Poisson brackets on \(T^*M\) does not depend on the choice of the
local coordinates on \(M\).
-
+
Theorem 3.3. Poisson bracket defines a structure of infinite-dimensional Lie
algebra5 on the space \(\mathcal {C}^\infty (T^*M)\) of smooth functions on the phase space.
5 The corresponding Lie group is the group of symplectomorphisms, a concept that will be introduced later.
@@ -3246,7 +3252,7 @@
-
+
Theorem 3.4. Let \(H=H(q,p)\) be the hamiltonian of an hamiltonian system (3.31) and \(F=F(q,p)\in \mathcal {C}^\infty (T^*M)\). Then, along the flow \((q(t), p(t))\) of \(H\), we have
-
+
Corollary 3.5. Given an hamiltonian system (3.31) with hamiltonian \(H\) and a smooth function \(F\) on its phase space, then \(F\) is a first integral if and only if \(H\) and \(F\) are in involution.
-
+
Corollary 3.6. The first integrals of a hamiltonian system form a subalgebra of the Lie algebra \(\mathcal {C}^\infty
(T^*M)\). Otherwise said, given two first integrals \(F\) and \(G\) of a hamiltonian system, then also the function \(\big \{F,G\big \}\) is a first integral of \(H\).
-
+
Corollary 3.7. The hamiltonian function \(H\) is a first integral of the hamiltonian system (3.31).
-
+
Corollary 3.8. The integral curves \(p = p(t)\) and \(q=q(t)\) of the hamiltonian system (3.31) belong to the levelset of the hamiltonian function, i.e.
@@ -3795,7 +3801,7 @@
-
+
Theorem 3.9. The solutions of a hamiltonian system (3.31) with hamiltonian \(H=H(q,p)\) are the critical points of the functional
-
+
Theorem 3.10. Given a function \(H=H(q,p)\) and a real number \(E\), then the (isoenergetic) trajectories
of the hamiltonian system with hamiltonian \(H\) are the critical points of the truncated action
-
+
Remark 3.5. Maupertuis principle is a variational principle in configuration space: it determines only the
trajectories of the motion, not the dynamics on the trajectories.
-
+
Example 3.7 (Geodesics and the free motion on riemannian
manifolds). Consider a riemannian manifold \((M, g)\), remember that the metric can be written as
-
+
Theorem 3.11. Given a riemannian manifold \((M,g)\), the trajectories of a free point particle on the manifold are the
geodesics of the metric.
-
+
Example 3.8 (Jacobi metric). Let’s consider a mechanical
system \((M,L)\), where \(L\) is a natural lagrangian for a particle of mass \(m\),
-
+
Theorem 3.12. The trajectories of the mechanical system \((M,L)\) in the domain (3.117) are the geodesic of the metric
-
+
Example 3.9 (Fermat principle and geometric optics).
Propagation of light in an isotropic medium is described by the hamiltonian
-
+
Theorem 3.13. Light in an isotropic medium propagates in a way that minimizes the travel time. The trajectories
followed by the light are the geodesics of the metric
@@ -4857,7 +4863,7 @@
-
+
Example 3.10. The transformation \(\Phi : (q,p) \mapsto (p, -q)\) is a canonical transformation. To
show it, it is enough to verify how the Poisson bracket acts on the coordinate functions. Let, as usual,
-
+
Example 3.11. The cotangent lift \(\Phi \) of a diffeomorphism \(\phi \)
defined on the base manifold \(M\), as defined below, is a canonical transformation.
-
+
Exercise 3.5. Consider the linear transformations on \(\mathbb {R}^{2n}\),
-
+
Theorem 3.14. Let \(\Phi : T^*M \to T^* M\) be a canonical transformation.
Then, every solution \(x = x(t)\) of the hamiltonian system with hamiltonian \(H\) is mapped by \(\Phi \) into a solution \(\Phi (x(t))\) of the hamiltonian system with hamiltonian \((\Phi ^{-1})^* H\).
@@ -5293,7 +5299,7 @@
-
+
Exercise 3.6. Let \(\Phi _t:T^*M \to T^*M\) denote a one–parameter
group of canonical transformations, i.e \(\big \{\Phi _t^* f, \Phi _t^* g\big \} = \Phi _t^*\big \{f,g\big \}\) for all \(t\in \mathbb {R}\). Show that the vector field \(X\) which generates the group,
6 Don’t let the \(T\)s scare you, here \(TT^*M\) simply means the tangent bundle to the manifold \(T^*M\).
@@ -5530,7 +5536,7 @@
-
+
Theorem 3.16. The hamiltonian vector field \(X_H\) associated to a smooth function
\(H\) on \(T^*M\) is an infinitesimal symmetry of Poisson bracket. Vice versa, for every infinitesimal symmetry \(X\) there exists a function \(H\) on \(T^*M\) such that \(X = X_H\).
-
+
Exercise 3.7. Let \(\Phi _t\) the one–parameter group of canonical transformations generated by the
quadratic hamiltonian
-
+
Exercise 3.8. Let \(\phi _t:M\to M\) denote a one–parameter group of diffeomorphisms of the base \(M\)
of the cotangent bundle \(T^*M\). Consider, as in Example 3.11, its cotangent lift
@@ -6078,7 +6084,7 @@
7 This should be compared with a symmetry of a lagrangian \(L:TM \to \mathbb {R}\), i.e. a diffeomorphism \(\phi : M \to M\) such that \(\phi ^* L = L\). Note that even though the formula looks very similar, the
symmetry of the hamiltonian is defined on the whole phase space and not induced by a symmetry on the base as in the Lagrangian case. In the hamiltonian formalism, position and momentum are really on equal footings.
@@ -6104,7 +6110,7 @@
-
+
Theorem 3.17 (Hamiltonian Noether theorem). Let \(H\) be a hamiltonian
on \(T^*M\) and \(F\) be a first integral of the hamiltonian system, then \(F\) generates a one–parameter group of symmetries of the hamiltonian system \(H\).
-
+
Corollary 3.18 (of Theorem 3.15). Let \(H, F \in \mathcal {C}^\infty (T^*M)\) be two hamiltonian functions in involution, i.e., \(\{H,F\}=0\). Then, their flows, respectively denoted here by \(\Phi _t^H\) and
\(\Phi _s^F\), commute:
@@ -6195,7 +6201,7 @@
-
+
Example 3.12. Consider \((q,p) = (x,y,z,p_x,p_y,p_z)\in T^* \mathbb {R}^3\) and the infinitesimal
generator \(M_z = x p_y - y p_x\) (which you may recognize as the \(z\)-component of the angular momentum).
-
+
Exercise 3.9. Let \(X(q), Y(q)\) denote two vector fields on \(M\) and define two hamiltonians linear in
the momenta as in (3.177):
-
+
Exercise 3.10. Given a mechanical system which is invariant with respect to space translations, show that
the components of the total momentum
@@ -6677,7 +6683,7 @@
8 I.e. \(\dd \omega = 0\)
-
+
Remark 3.6. The fact that \(\omega \) is closed and non-degenerate, in coordinates, can be translated into
-
+
Remark 3.7. Note that for an \(n\times n\) antisymmetric matrix \(A = - A^T\),
-
+
Lemma 3.19. The matrix of the two–form \(\omega \) defined by (3.201) in the coordinates \(q,p\) is given by
-
+
Theorem 3.20. The two–form \(\omega \) defined by (3.201) does not depend from the local choice of coordinates \(q^1, \ldots , q^n\) on the base \(M\) of the cotangent bundle \(T^* M\). This two–form is closed and non–degenerate,
therefore defining a symplectic structure on the manifold \(T^*M\).
@@ -6973,7 +6979,7 @@
-
+
Lemma 3.21. Let \(q^1, \ldots , q^n\) be a choice of local coordinates on a smooth manifold \(M\). Then the
tautological one–form
-
+
Remark 3.8 (A coordinate free explanation of the tautological
one–form). The name tautological comes from the physical intuition that velocity and momenta are necessarily proportional to one-another, but the connection between the two concepts runs deep as previous remarks have shown.
The tautological one-form enter into this picture as a foundational concept.
@@ -7352,7 +7358,7 @@
-
+
Exercise 3.11. Show that \(\big \{f,g\big \} = -\mathcal {L}_{X_f} g = \mathcal {L}_{X_g} f\),
where \(\mathcal {L}\) is the Lie derivative defined in (3.19). Remember that the inner product of a vector field with a function is zero by definition.
-
+
Exercise 3.12. Let \(X_H\) be an hamiltonian vector field on \(T^*M\).
Show that the following holds
@@ -7494,7 +7500,7 @@
-
+
Theorem 3.22. Let \(\Phi : T^*M \to T^*M\) be a diffeomorphism, then \(\Phi \) is a
canonical transformation if and only if it is a symplectomorphism.
-
+
Theorem 3.24. Let \(H\) be a function on the manifold \(T^*M\), then the hamiltonian flow \(\Phi _t\) generated by
\(H\) is a one–parameter group of symplectomorphisms of \((T^*M, \omega )\).
-
+
Theorem 3.25. Let \(X\) be an infinitesimal symplectomorphism on the symplectic manifold \((P,\omega )\), then
locally there exists a function \(H\) such that \(X = X_H\).
-
+
Example 3.13 (Time dependent hamiltonians). It is possible to use the symplectic formulation to describe hamiltonian systems with an explicit time dependence. To this end, one needs to consider the extended phase space \(T^*M\times \mathbb {R}^2\), as we have
already seen in Section 3.4.2.
@@ -7993,8 +7999,8 @@
@@ -8013,7 +8019,7 @@
-
+
Theorem 3.26 (Darboux’ Theorem). Let \((P, \omega )\) be a symplectic
manifold and \(\dim P = 2n\). Then, there exist local coordinates
@@ -8388,7 +8394,7 @@
-
+
Lemma 3.28. On a manifold \(M\) of dimension \(n\), let \(\omega \) be the symplectic form (3.201) on \(T^* M\). Then the exterior product of \(n\) copies of \(\omega \) is proportional to the volume form:
-
+
Exercise 3.13. Show that the definition of volume \(\Vol (D)\) presented above does not depend on the
choice of local coordinates \(q^1, \ldots , q^n\) on \(M\).
-
+
Theorem 3.29. Let \(\Phi :T^*M \to T^*M\) be a canonical transformation and
\(D\subset T^*M\) measurable, then
-
+
Theorem 3.30 (Liouville theorem). Let \(H\) be a hamiltonian function on
\(T^*M\). Then, the hamiltonian flow \(\Phi _t\) generated by \(H\) preserves the volume, that is, for every measurable subset \(D\subset T^*M\) and any \(t\) it holds
-
+
1.2 Hamilton’s variational principle
-
+
1.3 First steps with conserved quantities
-
+
1.4 Euler-Lagrange equations on smooth manifolds
-
+
2.2 The spherical pendulum
-
+
2.3 Intermezzo: small oscillations
-
+
2.4 Motion in a central potential
-
+
2.5 D’Alembert principle and systems with constraints
-
+
3.1 The Legendre Transform in the euclidean plane
-
+
3.2 Intermezzo: cotangent bundle and differential forms
-
+
3.3 The Legendre Transform
-
+
3.4 Poisson brackets and first integrals of motion
-
+
3.5 Variational principles of hamiltonian mechanics
-
+
3.6 Canonical transformations
-
+
3.7 The symplectic structure on the cotangent bundle
-
+
4.1 Lagrangian submanifolds
-
+
4.2 Canonical transformations revisited
-
+
4.3 Hamilton-Jacobi equations
-
+
4.4 The Liouville-Arnold theorem
-
+
5 Hamiltonian perturbation theory
-
+
5.1 Small oscillations revisited
-
+
5.3 A brief look at KAM theory
+
+Index
+
@@ -654,8 +660,8 @@
-
+
Theorem 4.1. The submanifold \(\Lambda \) defined by (4.1) is lagrangian if and only if the functions \(f_i\) are in involution on \(\Lambda \), i.e.
-
+
Example 4.1. One of the simplest example of lagrangian submanifolds are
-
+
Theorem 4.2. The submanifold \(\Lambda \) is lagrangian if and only if the following
equations are satisfied:
-
+
Theorem 4.3. Equations (4.12) define a lagrangian submanifold if and only if the following one–form on \(M\) is closed:
-
+
Remark 4.1. For a hamiltonian system with hamiltonian \(H\), we can compute
the generating function of a lagrangian submanifold from \(n\) first integrals \(f_1, \ldots , f_n\) in involution such that
@@ -1443,7 +1449,7 @@
-
+
Remark 4.2. If \(q^i = Q^i\), \(i=1,\ldots ,n\), then the function \(S=S(q)\). In fact, in this case,
(4.24) reads
@@ -1567,7 +1573,7 @@
-
+
Theorem 4.4. Let \(S_1(q,Q)\) be a function defined in a neighborhood of the point \((q_0,
Q_0)\) of two euclidean coordinate spaces. If
-
+
Remark 4.3. Not all canonical transformations are free: the identical transformation is not free, as \(q=Q\)
and thus they are not independent.
-
+
Example 4.2. Let \(\omega > 0\) be some fixed constant.
Consider the transformation \(\Phi \) given by
@@ -2047,7 +2053,7 @@
-
+
Example 4.3. Let \(Q=Q(q)\) be a change of coordinates on the base \(M\) of the cotangent bundle
\(T^*M\). This induces a transformation of the conjugate momenta \(p_i\mapsto P_i\) given by \(P_i = p_l \frac {\partial q^l}{\partial Q^i}\), and we already saw that \((q,p)\mapsto (Q,P)\) defines a canonical transformation. The
generating function of such transformation is given by
@@ -2119,7 +2125,7 @@
-
+
Theorem 4.5. Let \(\Phi :T^*M \to T^* M\) be the canonical transformation defined by the functions \(P(q,p)\) and
\(Q(q,p)\). Then, in a neighborhood of a point \((q_0, p_0)\) it is possible to choose one of the \(2^n\) collections of functions \((q, Q^{\vb * j}, P_{\vb * i})\) from (4.49) as independent coordinates, i.e., such that
@@ -2241,7 +2247,7 @@
-
+
Remark 4.4. Sometimes, in the literature you can find a different classification in the following terms:
@@ -2384,7 +2390,7 @@
-
+
Theorem 4.6. Every canonical transformation which is close to the identity admits a generating function of the form
-
+
Theorem 4.7. The change of coordinates (4.55) is an infinitesimal canonical transformation if and only if there exists a function \(K=K(q,p)\) such that
@@ -2726,7 +2732,7 @@
-
+
Theorem 4.8. Let \((q,p,t,E) \to (Q,P,T,\widetilde E)\) be a canonical transformation
in \(T^*M\times \mathbb {R}^2\). If
-
+
Example 4.4. Let \(H(q,p) = \frac {p^2}{2} + \alpha q\). Consider a time-dependent canonical
transformation \(\Phi _t\) generated by \(H\): let \(q(0) = Q\) and \(p(0) = P\) and
-
+
Theorem 4.9. The hamiltonian flow is a canonical transformation.
-
+
Remark 4.5. Choosing \((q,Q,t)\) as independent variables in the
proof above one gets
@@ -3561,7 +3567,7 @@
-
+
Theorem 4.10. Let \(H(q,p,t)\) be an hamiltonian function and \(S(q,b,t)\) be a complete integral of the
Hamilton-Jacobi equation (4.95) depending on \(n\) constants \(b=(b_1,\ldots ,b_n)\). If \(S\) satisfies the non–degeneracy condition
-
+
Theorem 4.11 (Liouville theorem). Let \(H(q,p)\) denote the hamiltonian
of a hamiltonian system with \(n\) degrees of freedom. Assume that there exist \(n\) first integral \(f_i\) of \(H\) which are in involution, i.e.
-
+
Example 4.5. Cyclic coordinates are an immediate example of separable coordinates. Assume for example
that \(q^1\) is cyclic, then the conjugate momentum \(p_1 = b_1\) is constant. The generating function has the form
-
+
Example 4.6. A simple example of separable system is given by a sum of hamiltonians depending each on
just one pair of conjugate coordinates:
-
+
Example 4.7 (Planar harmonic oscillator). Let’s reconsider an old friend of ours, the planar harmonic oscillator defined by a point particle of mass \(m\) attracted by an elastic force centered at \(0\in \mathbb {R}^2\) with stiffness \(k\).
-
+
Example 4.8. Let’s consider a more involved example of a point of
mass \(m\) moving in \(\mathbb {R}^3\) under the influence of an arbitrary potential \(U\).
-
+
Exercise 4.1. Show that the infinitesimal transformation
-
+
Exercise 4.2. Let \(Q = q^2 + \frac 12 \cos (q)\). Find \(P(q,p)\) such that the corresponding
coordinate transformation is canonical. Compute the corresponding generating function.
-
+
Exercise 4.3 (Parabolic coordinates). Let \((r,\phi ,z)\)
denote the cylindrical coordinates in \(\mathbb {R}^3\). Find the complete integral of the Hamilton-Jacobi equations for a point particle of mass \(m\) under the influence of the potential
-
+
Exercise 4.4 (Prolate spheroidal coordinates). Consider a
point particle of mass \(m\) under the gravitational attraction of two bodies which are fixed at the points \((d,0,0)\) and \((-d,0,0)\), \(d>0\). Show that the Hamilton-Jacobi equation is separable introducing the prolate spheroidal
coordinates obtained by
@@ -4675,7 +4681,7 @@
-
+
Exercise 4.5. Consider the motion of a free particle of mass \(m\) on a surface \(S\subset \mathbb
{R}^3\) parametrized by
-
+
Remark 4.6. In the integral (4.148) you can choose the initial point arbitrarily as long as it varies smoothly with respect to \(E\).
-
+
Theorem 4.12 (Liouville-Arnold).
Consider a completely integrable hamiltonian system on the phase space \(\mathbb {R}^{2n}\). Assume that there exists \(E^0 = (E^0_1, \ldots , E^0_n)\) such that the functions
-
+
Remark 4.7. A separable hamiltonian system is completely integrable. Indeed, for a separable system there
exists a canonical transformation \((q,p) \mapsto (Q,P)\) such that the hamiltonian \(H\) in the new coordinates does not depend on \(Q\), \(H=H(P)\). The functions \(P_1, \ldots , P_n\), which are in involution by construction, commute
with \(H\).
@@ -5126,7 +5132,7 @@
-
+
Lemma 4.13. Let \(M\) be a \(n\)-dimensional compact connected manifold. Assume that
at all points \(x\in M\) there exist \(n\) linearly independent vector fields \(X_1, \ldots , X_n\) which are pairwise commuting, i.e.,
-
+
Exercise 4.6. Show that the action \(\Phi _{\vb * t}\) is transitive, that is, for every two points \(x,
y\in M\), there exists \(\vb * t\in \mathbb {R}^n\) such that \(y = \Phi _{\vb * t} (x)\).
-
+
Exercise 4.7. Show that the subgroup \(\Gamma \) does not depend on \(x_0\). Hint: \(\Gamma \) is
transitive. Furthermore, \(\Gamma \) is a discrete subgroup of \(\mathbb {R}^n\), that is, there is an open set \(U \ni \vb * 0\) such that \(U\cap \Gamma = \{\vb * 0\}\).
-
+
Lemma 4.14. Let \(\Gamma \) be a discrete subgroup of \(\mathbb {R}^n\). Then, there
exist \(m\) linearly independent vectors \(e_1, \ldots , e_m \in \mathbb {R}^n\), \(0\leq m \leq n\), such that
-
+
Exercise 4.8. Extend the proof to completely integrable system on an arbitrary symplectic manifold.
(Solution: find the differences between our proof and [Kna18, Theorem 13.3]).
@@ -5871,7 +5877,7 @@
-
+
Theorem 4.15. Under the hypotheses of the Liouville-Arnold theorem (Theorem 4.12), for every compact connected component \(M_{E^0}\) there are a neighborhood \(M_{E^0}\subseteq U = U(M_{E^0}) \subseteq \mathbb {R}^{2n}\) and a symplectomorphism
-
+
Remark 4.8. Consider the (inverse) canonical transformation
-
+
Example 4.9 (Harmonic oscillator). Consider, once again,
the hamiltonian of a harmonic oscillator with mass \(m\) and stiffness \(k>0\):
-
+
Example 4.10 (Kepler problem – elliptic motion). Let’s
consider once again the Kepler problem for negative energies and, therefore, elliptic motions. For conciseness, we assume that the system is already reduced to the planar system. In this case, the invariant tori are two dimensional:
-
+
Exercise 4.9 (Anharmonic oscillator). Consider the
following hamiltonian system with one degree of freedom for a point particle of mass \(m\):
-
+
1.2 Hamilton’s variational principle
-
+
1.3 First steps with conserved quantities
-
+
1.4 Euler-Lagrange equations on smooth manifolds
-
+
2.2 The spherical pendulum
-
+
2.3 Intermezzo: small oscillations
-
+
2.4 Motion in a central potential
-
+
2.5 D’Alembert principle and systems with constraints
-
+
3.1 The Legendre Transform in the euclidean plane
-
+
3.2 Intermezzo: cotangent bundle and differential forms
-
+
3.3 The Legendre Transform
-
+
3.4 Poisson brackets and first integrals of motion
-
+
3.5 Variational principles of hamiltonian mechanics
-
+
3.6 Canonical transformations
-
+
3.7 The symplectic structure on the cotangent bundle
-
+
4.1 Lagrangian submanifolds
-
+
4.2 Canonical transformations revisited
-
+
4.3 Hamilton-Jacobi equations
-
+
4.4 The Liouville-Arnold theorem
-
+
5 Hamiltonian perturbation theory
-
+
5.1 Small oscillations revisited
-
+
5.3 A brief look at KAM theory
+
+Index
+
@@ -659,8 +665,8 @@
@@ -856,7 +862,7 @@
-
+
Lemma 5.1. Let \(A\) be a symmetric matrix and \(J\) be the symplectic matrix. The
characteristic polynomial of the matrix \(B=JA\) has the form
-
+
Theorem 5.2. Let \(H_2(x) = \frac 12\langle Ax,x\rangle \) be a generic quadratic hamiltonian. The the solution
\((q(t), p(t)) = (0,0)\) of the linear system (5.3) is stable if and only if all the roots of the polynomial \(P_n(z)\) from the Lemma 5.1 are real and negative.
@@ -993,7 +999,7 @@
-
+
Theorem 5.3. Let \(H(x) = \frac 12\langle Ax,x \rangle \) be a quadratic hamiltonian in \(\mathbb {R}^{2n}\)
such that the characteristic polynomial has the form
-
+
Example 5.1. If we consider the lagrangian
-
+
Remark 5.1. The hamiltonian systems with hamiltonian (5.13) is completely integrable. With the notation of (5.18), the invariant tori \(\mathbb {T}^n\) are defined by the
equations
@@ -1359,7 +1365,7 @@
-
+
Exercise 5.1 (The linear triatomic molecule). Use the
hamiltonian approach to small oscillations to compute the normal modes of the linear triatomic molecule from Section 2.3.3.
@@ -1573,7 +1579,7 @@
-
+
Example 5.2. Consider the following hamiltonian for a system with one degree of freedom
-
+
Exercise 5.2. Let \(H\) be an hamiltonian of the form
-
+
Theorem 5.4 (Birkhoff). Let \(H\) be an hamiltonian of the form
@@ -2441,7 +2447,7 @@
-
+
Theorem 5.5 (KAM theorem). Let \(H_0(I)\), \(I\in \mathcal
{I}\subset \mathbb {R}^n\), be a non–degenerate hamiltonian such that
-
+
Remark 5.2. Here, by being real analytic on \(\overline {\mathcal {I}}\times \mathbb {T}^n\) we
mean that the analyticity extends to a uniform neighborhood of \(\mathcal {I}\).
-
+
Remark 5.3. The hamiltonian nature of the equations is almost indispensable. Analogous results are true
for reversible systems, but in all cases the system has to be conservative: any kind of dissipation immediately destroys the Cantor family of tori, although isolated ones may persist as attractors.
@@ -2689,7 +2695,7 @@
-
+
Theorem 5.6 (Nekhoroshev). Let \(H_\epsilon \) be an hamiltonian of the
form (5.64) with \(H_0\) uniformly convex. Then there exists positive numbers \(a,b,\alpha ,\beta ,\epsilon _0\in \mathbb {R}_+\) such that for any \(|\epsilon
|< \epsilon _0\) and for any solution \((\phi (t), I(t))\) of Hamilton’s equations
diff --git a/chapter-6.html b/chapter-6.html
index 361c74c..22f53e0 100644
--- a/chapter-6.html
+++ b/chapter-6.html
@@ -234,7 +234,7 @@
-
+
@@ -278,179 +278,185 @@
-
+
1.2 Hamilton’s variational principle
-
+
1.3 First steps with conserved quantities
-
+
1.4 Euler-Lagrange equations on smooth manifolds
-
+
2.2 The spherical pendulum
-
+
2.3 Intermezzo: small oscillations
-
+
2.4 Motion in a central potential
-
+
2.5 D’Alembert principle and systems with constraints
-
+
3.1 The Legendre Transform in the euclidean plane
-
+
3.2 Intermezzo: cotangent bundle and differential forms
-
+
3.3 The Legendre Transform
-
+
3.4 Poisson brackets and first integrals of motion
-
+
3.5 Variational principles of hamiltonian mechanics
-
+
3.6 Canonical transformations
-
+
3.7 The symplectic structure on the cotangent bundle
-
+
4.1 Lagrangian submanifolds
-
+
4.2 Canonical transformations revisited
-
+
4.3 Hamilton-Jacobi equations
-
+
4.4 The Liouville-Arnold theorem
-
+
5 Hamiltonian perturbation theory
-
+
5.1 Small oscillations revisited
-
+
5.3 A brief look at KAM theory
+
+Index
+
@@ -660,8 +666,8 @@
+[Eur24] European Mathematical Society. “Elliptic Integral”. In: Encyclopedia of Mathematics. EMS Press, 2024. url: http://encyclopediaofmath.org/index.php?title=Elliptic_integral&oldid=46813.
+
[GF00] I.M. Gelfand and S.V. Fomin. Calculus of Variations. Dover Books on Mathematics. Dover Publications, New York, 2000. isbn: 9780486414485.
@@ -972,6 +988,224 @@
+A
+
+acceleration, section 1.1
+
+action, section 1.2
+
+action
+
+critical point, section 1.2
+
+B
+
+Banach space, section 1.2
+
+C
+
+configuration, section 1.1.1
+
+space, section 1.1.1
+
+D
+
+degrees of freedom, section 1.1.1
+
+diffemorphic, example 1.3
+
+E
+
+elliptic integrals, example 1.1
+
+equations of motion, section 1.1.1
+
+Euler-Lagrange equations, theorem 1.1
+
+F
+
+frames of reference, section 1.1
+
+Frechet
+
+derivative, section 1.2
+
+differentiable, section 1.2
+
+G
+
+generalized
+
+coordinates, section 1.1.1
+
+velocities, section 1.1.1
+
+gravitational constant, example 1.2
+
+H
+
+Hamilton’s principle, section 1.2
+
+I
+
+inertia, section 1.1
+
+L
+
+lagrangian, section 1.2
+
+N
+
+Newton
+
+principle of determinacy, section 1.1.1
+
+second law, section 1.1
+
+universal gravitation, example 1.2
+
+P
+
+pendulum, example 1.1
+
+ideal, example 1.1
+
+point particle, section 1.1
+
+system, section 1.1.1
+
+position, section 1.1
+
+principle of least action, section 1.2
+
+S
+
+spring, example 1.1
+
+constant, example 1.1
+
+Hooke’s law, example 1.1
+
+state space, section 1.1.1
+
+V
+
+variation, item 5
+
+velocity, section 1.1
+
-
-1.1.1 Motion in one degree of freedom
+
+1.1.1 From particles to systems of particles
-
+
1.2 Hamilton’s variational principle
-
+
1.2.1 Dynamics of point particles: from Lagrange back to Newton
-
+
1.3 First steps with conserved quantities
-
+
1.3.1 Back to one degree of freedom
-
+
1.3.2 The conservation of energy
-
+
1.3.3 Fun with the phase portrait
-
+
1.4 Euler-Lagrange equations on smooth manifolds
-
+
2.1.1 Homogeneity of space: total momentum
-
+
2.1.2 Isotropy of space: angular momentum
-
+
2.1.3 Homogeneity of time: the energy
-
+
2.1.4 Scale invariance: Kepler’s third law
-
+
2.2 The spherical pendulum
-
+
2.3 Intermezzo: small oscillations
-
+
2.3.1 Decomposition into characteristic oscillations
-
+
2.3.3 The linear triatomic molecule
-
+
2.4 Motion in a central potential
-
+
2.5 D’Alembert principle and systems with constraints
-
+
3.1 The Legendre Transform in the euclidean plane
-
+
3.2 Intermezzo: cotangent bundle and differential forms
-
+
3.3 The Legendre Transform
-
+
3.4 Poisson brackets and first integrals of motion
-
+
3.4.1 The symplectic matrix
-
+
3.4.2 A brief detour on time-dependent hamiltonians
-
+
3.5 Variational principles of hamiltonian mechanics
-
+
3.6 Canonical transformations
-
+
3.6.1 Hamiltonian vector fields
-
+
3.6.2 The hamiltonian Noether theorem
-
+
3.7 The symplectic structure on the cotangent bundle
-
+
3.7.1 Symplectomorphisms and generating functions
-
+
4.1 Lagrangian submanifolds
-
+
4.2 Canonical transformations revisited
-
+
4.2.1 The generating function \(S_1\)
-
+
4.2.2 The generating functions \(S_2\) and \(S_3\)
-
+
4.2.3 Infinitesimal canonical transformations
-
+
4.2.4 Time-dependent hamiltonian systems
-
+
4.3 Hamilton-Jacobi equations
-
+
4.3.1 Separable hamiltonians
-
+
4.4 The Liouville-Arnold theorem
-
+
4.4.1 Action-angle variables
-
+
5 Hamiltonian perturbation theory
-
+
5.1 Small oscillations revisited
-
+
5.3 A brief look at KAM theory
+
+Index
+2.5 D’Alembert
2.5 D’Alembert
2.5 D’Alembert
2.5 D’Alembert
2.5 D’Alembert
2.5 D’Alembert
2.5 D’Alembert
2.5 D’Alembert
Hamiltonian Mechanics
-3 Hamiltonian mechanics
-
+3 Hamiltonian mechanics
+
3 Hamiltonian mec
-
3.1 The Legendre Transform in the euclidean plane
-
+3.1 The Legendre Transform in the euclidean plane
+
3.1 The Legendre
3.1 The Legendre
-
3.2 Intermezzo: cotangent bundle and differential forms
-
+3.2 Intermezzo: cotangent bundle and differential forms
+
3.2 Intermezzo: c
-
3.3 The Legendre Transform
-
+3.3 The Legendre Transform
+
3.3 The Legendre
3.3 The Legendre
@@ -1426,7 +1432,7 @@
3.3 The Legendre
>
3.3 The Legendre
3.3 The Legendre
3.3 The Legendre
3.3 The Legendre
3.3 The Legendre
\begin{align}
-\mathbb {F}L(v) w := \frac {\dd }{\dd s} L(q, v + s w) \big |_{s=0}.
+\mathbb {F}L(v) w := \dv {s} L(q, v + s w) \big |_{s=0}.
\end{align}
That is, \(\mathbb {F}L(v) w\) is the derivative of \(L\) at \(v\) in the direction \(w\) along the fiber \(T_qM\). The last observation is crucial: the map is fiber-preserving in the sense that it maps the fiber \(T_qM\) to the fiber \(T^*_qM\). If we
denote local coordinates on \(M\) by \(q=(q^1, \ldots , q^n)\), vectors on \(T_qM\) take the form \(\dot q = \dot q^i \frac {\partial }{\partial q^i}\big |_q \simeq (\dot q^1, \ldots , \dot q^n)\) and the fiber derivative can be
@@ -1653,7 +1659,7 @@
3.3 The Legendre
3.3 The Legendre
@@ -1808,7 +1814,7 @@
3.3 The Legendre
3.3 The Legendre
3.3 The Legendre
3.3 The Legendre
3.3 The Legendre
3.3 The Legendre
3.3 The Legendre
3.3 The Legendre
3.3 The Legendre
3.3 The Legendre
3.3 The Legendre
3.3 The Legendre
-
3.4 Poisson brackets and first integrals of motion
-
+3.4 Poisson brackets and first integrals of motion
+
@@ -3074,7 +3080,7 @@ 3.4 Poisson brack
3.4 Poisson brack
3.4 Poisson brack
3.4 Poisson brack
3.4 Poisson brack
3.4 Poisson brack
3.4 Poisson brack
@@ -3381,7 +3387,7 @@
3.4 Poisson brack
3.4 Poisson brack
3.4 Poisson brack
-
3.4.1 The symplectic matrix
-
+3.4.1 The symplectic matrix
+
@@ -3606,8 +3612,8 @@ 3.4.1 The symplec
-
3.4.2 A brief detour on time-dependent hamiltonians
-
+3.4.2 A brief detour on time-dependent hamiltonians
+
@@ -3775,8 +3781,8 @@ 3.4.2 A brief det
-
3.5 Variational principles of hamiltonian mechanics
-
+3.5 Variational principles of hamiltonian mechanics
+
3.5 Variational p
3.5 Variational p
@@ -3931,7 +3937,7 @@
3.5 Variational p
3.5 Variational p
3.5 Variational p
@@ -4144,7 +4150,7 @@
3.5 Variational p
3.5 Variational p
3.5 Variational p
3.5 Variational p
3.5 Variational p
@@ -4544,7 +4550,7 @@
3.5 Variational p
3.5 Variational p
3.5 Variational p
-
3.6 Canonical transformations
-
+3.6 Canonical transformations
+
3.6 Canonical tra
3.6 Canonical tra
3.6 Canonical tra
3.6 Canonical tra
3.6 Canonical tra
@@ -5233,8 +5239,8 @@
3.6 Canonical tra
-
3.6.1 Hamiltonian vector fields
-
+3.6.1 Hamiltonian vector fields
+
3.6.1 Hamiltonian
3.6.1 Hamiltonian
3.6.1 Hamiltonian
@@ -5614,7 +5620,7 @@
3.6.1 Hamiltonian
3.6.1 Hamiltonian
@@ -5916,7 +5922,7 @@
3.6.1 Hamiltonian
3.6.1 Hamiltonian
3.6.1 Hamiltonian
-
3.6.2 The hamiltonian Noether theorem
-
+3.6.2 The hamiltonian Noether theorem
+
3.6.2 The hamilto
3.6.2 The hamilto
3.6.2 The hamilto
@@ -6139,7 +6145,7 @@
3.6.2 The hamilto
3.6.2 The hamilto
3.6.2 The hamilto
3.6.2 The hamilto
3.6.2 The hamilto
-
3.7 The symplectic structure on the cotangent bundle
-
+3.7 The symplectic structure on the cotangent bundle
+
3.7 The symplecti
3.7 The symplecti
3.7 The symplecti
3.7 The symplecti
3.7 The symplecti
@@ -6950,7 +6956,7 @@
3.7 The symplecti
3.7 The symplecti
3.7 The symplecti
@@ -7133,7 +7139,7 @@
3.7 The symplecti
3.7 The symplecti
3.7 The symplecti
3.7 The symplecti
-
3.7.1 Symplectomorphisms and generating functions
-
+3.7.1 Symplectomorphisms and generating functions
+
3.7.1 Symplectomo
3.7.1 Symplectomo
@@ -7794,7 +7800,7 @@
3.7.1 Symplectomo
3.7.1 Symplectomo
3.7.1 Symplectomo
3.7.1 Symplectomo
-
3.7.2 Darboux Theorem
-
+3.7.2 Darboux Theorem
+
3.7.2 Darboux The
3.7.2 Darboux The
@@ -8372,8 +8378,8 @@
3.7.2 Darboux The
-
3.7.3 Liouville theorem
-
+3.7.3 Liouville theorem
+
3.7.3 Liouville t
3.7.3 Liouville t
@@ -8503,7 +8509,7 @@
3.7.3 Liouville t
3.7.3 Liouville t
3.7.3 Liouville t
@@ -8612,7 +8618,7 @@
3.7.3 Liouville t
Hamiltonian Mechanics
-4 Integrable systems
-
+4 Integrable systems
+
4 Integrable syst
-
4.1 Lagrangian submanifolds
-
+4.1 Lagrangian submanifolds
+
@@ -715,7 +721,7 @@ 4.1 Lagrangian su
4.1 Lagrangian su
@@ -891,7 +897,7 @@
4.1 Lagrangian su
4.1 Lagrangian su
4.1 Lagrangian su
@@ -1073,7 +1079,7 @@
4.1 Lagrangian su
4.1 Lagrangian su
@@ -1210,7 +1216,7 @@
4.1 Lagrangian su
4.1 Lagrangian su
-
4.2 Canonical transformations revisited
-
+4.2 Canonical transformations revisited
+
4.2 Canonical tra
4.2 Canonical tra
-
4.2.1 The generating function \(S_1\)
-
+4.2.1 The generating function \(S_1\)
+
4.2.1 The generat
4.2.1 The generat
@@ -1728,7 +1734,7 @@
4.2.1 The generat
4.2.1 The generat
4.2.1 The generat
-
4.2.2 The generating functions \(S_2\) and \(S_3\)
-
+4.2.2 The generating functions \(S_2\) and \(S_3\)
+
4.2.2 The generat
4.2.2 The generat
4.2.2 The generat
@@ -2292,7 +2298,7 @@
4.2.2 The generat
@@ -2344,8 +2350,8 @@
4.2.2 The generat
-
4.2.3 Infinitesimal canonical transformations
-
+4.2.3 Infinitesimal canonical transformations
+
4.2.3 Infinitesim
4.2.3 Infinitesim
4.2.3 Infinitesim
@@ -2661,8 +2667,8 @@
4.2.3 Infinitesim
-
4.2.4 Time-dependent hamiltonian systems
-
+4.2.4 Time-dependent hamiltonian systems
+
4.2.4 Time-depend
4.2.4 Time-depend
@@ -2913,7 +2919,7 @@
4.2.4 Time-depend
4.2.4 Time-depend
4.2.4 Time-depend
@@ -3238,7 +3244,7 @@
4.2.4 Time-depend
4.2.4 Time-depend
-
4.3 Hamilton-Jacobi equations
-
+4.3 Hamilton-Jacobi equations
+
4.3 Hamilton-Jaco
4.3 Hamilton-Jaco
4.3 Hamilton-Jaco
-
4.3.1 Separable hamiltonians
-
+4.3.1 Separable hamiltonians
+
@@ -3788,7 +3794,7 @@ 4.3.1 Separable h
4.3.1 Separable h
4.3.1 Separable h
4.3.1 Separable h
4.3.1 Separable h
-
4.3.2 A few exercises
-
+4.3.2 A few exercises
+
@@ -4458,7 +4464,7 @@
4.3.2 A few exerc
4.3.2 A few exerc
4.3.2 A few exerc
4.3.2 A few exerc
4.3.2 A few exerc
4.3.2 A few exerc
-
4.4 The Liouville-Arnold theorem
-
+4.4 The Liouville-Arnold theorem
+
@@ -4822,7 +4828,7 @@ 4.4 The Liouville
4.4 The Liouville
4.4 The Liouville
4.4 The Liouville
@@ -5375,7 +5381,7 @@
4.4 The Liouville
4.4 The Liouville
@@ -5508,7 +5514,7 @@
4.4 The Liouville
4.4 The Liouville
4.4 The Liouville
4.4 The Liouville
4.4 The Liouville
-
4.4.1 Action-angle variables
-
+4.4.1 Action-angle variables
+
4.4.1 Action-angl
4.4.1 Action-angl
@@ -6236,7 +6242,7 @@
4.4.1 Action-angl
4.4.1 Action-angl
4.4.1 Action-angl
4.4.1 Action-angl
Hamiltonian Mechanics
-5 Hamiltonian perturbation theory
-
+5 Hamiltonian perturbation theory
+
5 Hamiltonian per
-
5.1 Small oscillations revisited
-
+5.1 Small oscillations revisited
+
5.1 Small oscilla
5.1 Small oscilla
@@ -970,7 +976,7 @@
5.1 Small oscilla
5.1 Small oscilla
5.1 Small oscilla
5.1 Small oscilla
5.1 Small oscilla
5.1 Small oscilla
-
5.2 Birkhoff normal forms
-
+5.2 Birkhoff normal forms
+
5.2 Birkhoff norm
5.2 Birkhoff norm
5.2 Birkhoff norm
5.2 Birkhoff norm
-
5.3 A brief look at KAM theory
-
+5.3 A brief look at KAM theory
+
5.3 A brief look
5.3 A brief look
5.3 A brief look
5.3 A brief look
-
5.4 Nekhoroshev theorem
-
+5.4 Nekhoroshev theorem
+
5.4 Nekhoroshev t
Hamiltonian Mechanics
-6 Conclusion
-
+6 Conclusion
+
6 Conclusion
-Bibliography
-
+Bibliography
+
@@ -792,6 +798,16 @@
+
+Bibliography
Bibliography
Index
+
+
+
+Contents
