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

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.

newton, animation, semi-implicit euler method, gravity

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

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

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

animation, ode, explicit euler method

Simulates the simple pendulum and damped simple pendulum

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

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

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

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

einstein, animation, brown, diffusion, random walk

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

animation, schrödinger equation, tunneling, scattering

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

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

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

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

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