# Symplectic Numerical Integration

## Contents

## Introduction

In the IB and II Numerical Analysis Courses, we are concerned with the accuracy of solutions and related issues: how fast does the error decay? **However**, many equations in practice have special interesting properties: think of evolutions constrained to surfaces, or in physics, of conservation of energy. It makes sense to investigate how these properties are handled by various methods!

## Preserving Phase-plane area

We consider the simple case of the ODE : the *non-linear pendulum*. Recall the concept of the **phase-plane** from **IA Differential Equations**: this is a plot of against .

A map from the phase-plane to itself is said to be **symplectic** if it preserves areas. The **flow-map** of the ODE above, which evolves the system from via the ODE, is one example of a symplectic map. It is not unreasonable then, to consider numerical methods inducing symplectic maps.

There is also a connection to conservation of energy. However, the 'energy' conserved by the symplectic numerical method will in general be different from the energy of the original system. But for a 'good' numerical method, the perturbed energy should be 'close'.

To show these concepts in action, we have produced an animation of a disc in the phase plane evolving under the ODE above. The numerical solution is produced by the **Symplectic Euler (SE)** [1] solver:

Although the evolution contorts the disc, it preserves the disc's original area.

## Comparison (Short-term)

How does SE fare against other solvers, say the now-familiar **(Explicit/Forward) Euler (EE)** method or the built-in solver `ode45`? The following animation demonstrates.

The evolution should be along an ellipse, and it's clear that EE goes completely astray. SE and `ode45` fare better. Note that the ellipse-shaped orbit for SE is slightly rotated relative to that for `ode45`. It is also rotated relative to the exact trajectory - remember that symplectic methods conserve the energy of a modified system.

## Comparison (Long-term)

In some sense, the previous comparison is not terribly fair: SE is a first-order method while `ode45` is a high-order Runge-Kutta method with error control! However, there is a sense in which SE still does better: conservation of energy. This requires a very long-term investigation: we just plot the result below as a single image: (Program symplectic_test_long_time_behaviour.m generates the following image WARNING: it takes quite a long time)

The thick red ellipse is the result of the trajectory of the `ode45` solution slowly spiralling in towards the origin. It should not do this! The SE trajectory, however, remains thin.

## Further Reading

[2] is a good, if obvious place to start. Chapter 5 of [3] is an excellent (if purer) introduction to the main themes of numerical methods that preserve special properties of equations. Another symplectic integrator is [4].

## References

[1] http://en.wikipedia.org/wiki/Symplectic_Euler_method

[2] http://en.wikipedia.org/wiki/Symplectic_integrator

[3] A First Course in the Numerical Analysis of Differential Equations; Arieh Iserles

[4] http://en.wikipedia.org/wiki/Verlet_integration

## Code

All files as zip archive: symplectic_integrators_all.zip

`WARNING!` The `avi` files produced are quite large.