Pendulum Exercise

Solving differential equations on the computer demands that you approximate the continuous-time evolution with a discrete time step. Accuracy, stability, and fidelity are the three criteria for a good algorithm. We study three methods used in different contexts for solving the problem of a simple pendulum.

The Euler method, very crude and unstable for simple systems, is actually often used in solving nonlinear partial-differential equations.
The Verlet method is used in the study of Hamiltonian dynamical systems. Fidelity is its strong point: it is not as accurate as more sophisticated methods, but the dynamics is an exact solution (up to rounding errors) of an approximate Hamiltonian: it retains volume in phase space, and exactly conserves an approximation to the energy.
Sophisticated packages for adaptively solving differential equations, which switch between different orders and between explicit and implicit methods, are appropriate for most small systems of ordinary differential equations.

Links

James P. Sethna, Christopher R. Myers.

Last modified: August 24, 2006

Statistical Mechanics: Entropy, Order Parameters, and Complexity, now available at Oxford University Press (USA, Europe).