Lattice (Oscillators on String) Simulator

Numerically simulated program that emulates the famous experiment conducted in 1953 by Enrico Fermi, John Pasta, Stanislaw Ulam, and Mary Tsingou. The program will solve a dual system of ODEs and find displacements of oscillators in a 1D lattice with user-specified parameters, display an animation of the oscillators' displacements through time and finally perform a Fourier analysis.

## Video Demonstration (Non-Linear Case)

How to run the program

• Download or clone repository
• Run view.py
• In the Chrome browser, navigate to http://127.0.0.1:3000/
• Fill in the HTML form and submit
• Wait for data to be processed, then watch the Matplotlib-generated animated solution

Simulation Program Structure

HTML GUI

Two HTML forms

Form 1

• Has every input apart from initial conditions, x(t0) for all i.
• Hides submit button until every input field contains a number.

Form 2

• First asks what shape the string will take initially. Options are limited to sine, half-sine or parabola (since x0 and xL must be zero).
• If user inputs "sine" or "half-sine", ask for amplitude and number of wavelengths or half-wavelengths, resp. Check no. of wavelengths is an integer.
• If user inputs "parabola", ask for stationary point height.
  • Extra feature: Add another option: "polynomial":
    • Allow user to chooose polynomial degree and coefficients
      • i.e. if degree is "4", ask user for five coefficients (in order)
    • allow user to click button to randomize coefficients before/after choosing polynomial degree
    • keep button shaded and unclickable until polynomial satisfies the constraints of x0 = xL = 0
      • or, if polynomial doesn't fit constraints, translate it until it does without changing (challenging)

Python integration

• Take data from user and call main "simulate()" function with paramaters
• Represent processed data and solution on new HTML page to show user

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Solve System of ODEs

Simultaneous Equations

• dx_i/dt for every oscillator i, i.e. N simultaneous equations

Perform RungeKutta 4th-order method on d^2x/dt^2 to determine discretised velocities

• Create array of arrays, where each array holds all velocity values for one of N oscillators

Repeat process for dx/dt for displacements

• Instead of using a derivative function, we'll have to index the velocity arrrays to find dx_i/dt at each time-step

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Plot x(t, i)

• X-axis will be the distance along string (i-value)
• To show the progression through time, we will either animate the graph or plot M graphs in one figure, where M is the number of time-steps between t_0 and t_final
  • Matplotlib has an animation class

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Analyse the change in the first few Fourier Series coefficients over time in the non-linear case

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Additional Features - Ideas

• If user inputs any parameters that may cause instability, kill the program and print error or prevent user from submitting the Parameters form in this case.

• Create a range of preset polynomials that satisfy the boundary conditions for the user to choose from when setting initial conditions. Create a button that when clicked displays a subsection of polynomial graphs that can be selected and simulated.