From 3400918805e5f301f03c957a47bf1943c54c7e5e Mon Sep 17 00:00:00 2001 From: Marcello Seri Date: Sat, 21 Sep 2024 15:42:44 +0200 Subject: [PATCH] Fixes, including disappearing arguments Signed-off-by: Marcello Seri --- hm.tex | 20 ++++++++++---------- images/lissajous.pdf | Bin 12124 -> 32913 bytes 2 files changed, 10 insertions(+), 10 deletions(-) diff --git a/hm.tex b/hm.tex index 79010e0..edb1295 100644 --- a/hm.tex +++ b/hm.tex @@ -77,6 +77,7 @@ \let\forlistloop\etoolboxforlistloop \begin{warpprint} + \usepackage{picins} %\usepackage[numbers, sort]{natbib} %\setlength{\bibsep}{3pt} %\renewcommand{\bibfont}{\small} @@ -860,12 +861,12 @@ \chapter*{Preface} \begin{remark} Often in physics, the kinetic energy is instead defined as \emph{the work required to bring the mass from rest to some speed~$\dot{\vb{x}}$}. One can then perform the computation above backwards - \begin{equation} + \begin{align} \int_{x_0}^{x^1} \left\langle\vb*{F}(\vb*{x}), \dd{\vb*{x}}\right\rangle - = \int_{t_0}^{t_1} \left\langle\dot{\vb*{x}}(t), \vb*{F}(\vb*{x}(t))\right\rangle\; \dd t - = \int_{t_0}^{t_1} m \langle\dot{\vb*{x}}(t), \ddot{\vb*{x}}(t)\rangle\; \dd t + &= \int_{t_0}^{t_1} \left\langle\dot{\vb*{x}}(t), \vb*{F}(\vb*{x}(t))\right\rangle\; \dd t \\ + &= \int_{t_0}^{t_1} m \langle\dot{\vb*{x}}(t), \ddot{\vb*{x}}(t)\rangle\; \dd t = \int_{t_0}^{t_1} \dv{t} \frac{m}{2} \langle\dot{\vb*{x}}(t), \dot{\vb*{x}}(t)\rangle \; \dd t, - \end{equation}, + \end{align}, leading to the definition of kinetic energy as $T = \frac{m}{2} \|\dot{\vb{x}}\|^2$. \end{remark} @@ -1239,9 +1240,8 @@ \chapter*{Preface} \begin{example}[Double pendulum]\label{ex:2pendulum} 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. \warpprintonly{ - \InsertBoxR{0}{ - \includegraphics[width=0.3\textwidth]{images/HM-1-12.pdf} - }% + \parpic[r]{\includegraphics[width=0.3\textwidth]{images/HM-1-12.pdf}} + % } \warpHTMLonly{ \begin{figure}[htbp] @@ -1320,7 +1320,7 @@ \chapter*{Preface} \begin{figure}[ht] \centering - \includegraphics[width=.75\linewidth,trim={0 50pt 0 50pt},clip]{images/lissajous.pdf} + \includegraphics[width=.75\linewidth]{images/lissajous.pdf} \caption{Some casually selected Lissajous figures. The top right example (in orange) and the one in the center at the bottom (in purple) are degenerate closed curves squeezed to a line.} \label{img:lissajous} \end{figure} @@ -1448,11 +1448,11 @@ \chapter*{Preface} 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 $q$. The most immediate way to compute the components $\dot q^i$ is by using the definition of tangent vectors as class of equivalence of tangents to curves \cite[Chapter 2.5]{lectures:aom:seri}. - Let $\gamma:\mathbb{R} \to M$ be a curve such that $\gamma(0) = q$ and $\dv{\gamma}{t}|_{t=0} = v$, then $\dot q^i = \dv{}{t}(q^i\circ\gamma)|_{t=0}$ is given by the $i$th component of $\dv{}{t}(\phi\circ\gamma)|_{t=0}$. + Let $\gamma:\mathbb{R} \to M$ be a curve such that $\gamma(0) = q$ and $\dv{\gamma}{t}|_{t=0} = v$, then $\dot q^i = \dv{}{t}\/(q^i\circ\gamma)|_{t=0}$ is given by the $i$th component of $\dv{}{t}\/(\phi\circ\gamma)|_{t=0}$. The notation used for the basis versors of the tangent space in coordinates is not casual: in differential geometry tangent vectors are also \emph{derivations}. If $f\in C^\infty(M)$, the space of smooth functions from $M$ to $\mathbb{R}$, one can compute that for any tangent vector $v\in T_qM$, \begin{equation} - v(f)=\dv{}{t}(f\circ\gamma)\big|_{t=0}, \qquad \gamma(0) = q, \; \dot \gamma(0) = v. + v(f)=\dv{}{t}\/(f\circ\gamma) \big|_{t=0}, \qquad \gamma(0) = q, \; \dot \gamma(0) = v. \end{equation} In local coordinates this corresponds to computing the directional derivative of $f$ in the direction of $v$ at the point $q$: i.e. $v(f)$ can be identified with $\langle Df(q),v\rangle$. diff --git a/images/lissajous.pdf b/images/lissajous.pdf index ef1d23e64f76d0459797e89d3fe874d82e0c5943..0400c25574ed3c49f7f7996b713479bfd9e0593c 100644 GIT binary patch delta 31203 zcmZ6xb8u!&)b1VIwrx#pOmK%2n|Eycj_pirPi)(^HL)@A#5wc4^?vo8I#s*6y8q~1 ztAE$pUESB(TLy`N5D5U#ETg0f)~Ja2*sRr0Z7rGP1ZaXH8tw^s0=We|U|oh6r^EN@ z_*EauagB9F6_bP2px%J$28}40@r}&2{F5&HZ^@aVKAk7N*MEPz@c-4T3tf~O`SmDD zleH=#*6Tywa9RT@K*vSlKZ|Av;}#YWP*}coQi?rtfUVMdci&wtC>&40@D7OZE)k*o$vF 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