Stabilising an Inverted Pendulum on a Cart

animation, 4th order runge-kutta, system of equations

The 4th order Runge-Kutta method was used to integrate the equations of motion for the system, then the pendulum was stabilised on its inverted equilibrium point using a proportional gain controller and linear quadratic regulator.

  Gravity Assist

animation, gravity, newton, semi-implicit euler method

Explaining the concept and simulating gravitational slingshot of a spacecraft passing a planet.

  Double Pendulum and Chaos

animation, differential equation, Lagrangian, Euler-Lagrange equations, chaos, phase space, odeint

Discusses the chaotic motion of the double pendulum using a phase-space diagram

  Simple Pendulum

animation, explicit euler method, ode

Simulates the simple pendulum and damped simple pendulum

  Roller Coaster

animation, gravity, newton, 4th order runge-kutta, interpolation, cubic splines

The motion of a rolling object on an arbitrary track is analyzed.

  Solving the Time-Dependent Schrödinger equation

animation, eigenenergy, eigenstate, schrödinger equation, tunneling, scattering, ehrenfest's theorem

The Time-Dependent Schrödinger equation is solved by expressing the solution as a linear combination of (stationary) solutions of the Time-Independent Schrödinger equation.

  Introduction to Brownian Motion and Diffusion

animation, diffusion, einstein, random walk, brown

A brief introduction to Brownian motion and its connection with diffusion. A system of Brownian particles in 2D is simulated and visualised.

  One-Dimensional Wave Propagation

animation, schrödinger equation, tunneling, scattering

A one-dimensional wave-packet is propagated forward in time for various different potentials.

  Planetary Motion - Three Body Problem

animation, space, gravity, newton, embedded runge-kutta pair, angular momentum

Applying the fourth order Runge-Kutta method and the adaptive step size Runge-Kutta method to calculate the trajectories of three bodies.

  Partial Differential Equations - Two Examples

animation, laplace's equation, finite-differences, pde, differential equation, stability, implicit euler method

This module shows two examples of how to discretize partial differential equations: the 2D Laplace equation and 1D heat equation.