18. Sep. 2023: First Lecture

25. Sep. 2023: Newton's Method and Kepler's Equation

02. Oct. 2023: Population Growth, Chaos and Fractals

09. Oct. 2023: Fractals using Complex Numbers, start of ODEs

16. Oct. 2023: Ordinary Differential Equations, Lotka-Voltera System

23. Oct. 2023: Symplectic Integrators, Simple Pendulum

30. Oct. 2023: The N-body Problem, Simulating the Solar System

Your solutions should be handed in 7 days after each assignment has been given, i.e. Sunday night by 21:00 one week after the Monday lecture. Assignments should be individual and should be in python and provide a correct virtual environment! (if you stick to standard libraries like numpy, matplotlib, scipy… you can also just submit your python source code together with the output of your program)

For help getting started with virtual environments, please read carefully Python Virtual Environments for Pip and Python Virtual Environments for Conda.

Please hand in the following in Teams:

1. The working python source code
2. The requirements.txt file for your virtual environment (if you used a virtual environment)
3. A .pdf or .png image or animation of the output of your program

Template: template.zip

Instructions:

Please add the names of the people you work together (if you do) to the comment section of your python scripts.

Create a virtual environment using

Pip

- run virtualenv yourenv_name to create a virtual environment

- run source yourenv_name/bin/activate to activate yourenv_name

- install necessary libraries that you want using pip install package_name

- work in that directory, get your outputs (*.pdf, *.png, *jpeg, *.mp4, etc…)

- run pip freeze > requirements.txt to get your list of libraries

Conda

- run conda create -n yourenvname python=x.x anaconda to create a virtual environment

- run source activate yourenvname to activate yourenv_name

- install necessary libraries that you want using conda install -n yourenv_name package_name

- work in that directory, get your outputs (*.pdf, *.png, *jpeg, *.mp4, etc…)

- run conda list –export > requirements.txt to get your list of libraries

1. Implement the bisection root finding method presented in the first lecture for a function f of your choice (e.g. x^x - 100 = 0 or the quadratic function f(x) = ax² + bx + c, for which you can compare to the analytical result)! (to submit by 24 September 2023, 9pm)
2. Plot (and/or animate) the elliptical orbit of a planet around the sun by repeatedly solving Kepler's equation with Newton's method (or the bisection method), as explained in the lecture! (to submit by 01 October 2023, 9pm)
3. Draw a Feigenbaum diagram that results from solving the logistic equation. (Optional: Implement a function that allows you to zoom into the Feigenbaum diagram) (to submit by 8 October 2023, 9pm)
4. Draw some Julia sets with various constants c. You can start with the Mandelbrot set as it was explained in the lecture and the exercise class! (to submit by 15 October 2023, 9pm)
5. Ordinary Differential Equations: Solve the Lotka-Volterra equation using the Euler method and the midpoint Runge-Kutta method (optional: 4th order Runge Kutta method) and compare the results. Make two plots: the time dependence of both populations (mice and foxes), and the phase diagram, using different initial conditions (to submit by 22 October 2023, 9pm)
6. Symplectic Integrators: Use the Leap-Frog method to make a phase plot (p vs q) of the simple pendulum for different total energies. Compare the results with what you get using the Forward Euler method and the midpoint Runge-Kutta method. (to submit by 29 October 2023, 9pm).
7. Make a solar system orrery following the steps outlined in the lecture. Also plot the path of the solar system center of mass.(to submit by 5 November 2023, 9pm)
8. Elliptical partial differential equations: Solve the Poisson equation for the electromagnetic potential using the SOR method described in the lecture, with boundary conditions given by a 1000 Volt stick in the center of a 0 Volt box (as depicted in the lecture notes). Plot the contours of the resulting potential. (to submit by 12 November 2023, 9pm)