## Numerical Hamiltonian Problems

Hamiltonian systems, given by

where H(p,q) a Hamiltonian function and d is the number of degrees
of
freedom of the system can be seen to possess two remarkable properties:

- The solutions preserve the Hamiltonian H(p,q), i.e., H=Constant
- The corresponding flow is sympletic, i.e., preserves the
differential
2-form
. For d=1, this corresponds to
preservation of
area.

Both of the above properties are usually destroyed by numerical
methods
applied to Hamiltonian systems. A class of numerical methods, referred
to as
sympletic methods, do produce numerical solutions with the above
mentioned two
properties. Sympletic Integration methods are the key to the successful
long
time integration of Hamiltonian Systems.

The image on this page is the exact solution to Kepler's Orbit
problem in
black, with the corresponding numerical solution produced by a
non-sympletic
fourth order explicit Runge-Kutta method. A fourth order sympletic
method will
produce dramatically different results.

The Applet
demonstrates the integration of six model Hamiltonian
systems by a variety of sympletic and non-sympletic methods. The
methods are
listed below with their order of accuracy in parentheses and with *
denoting
that the methods is sympletic.

__Explicit Methods:__ Euler(1), Improved Euler(2),
Runge-Kutta(3),
"Classical" Runge-Kutta(4), Runge-Kutta(6), Stormer-Verlet
Runge-Kutta-Nystrom(2)*, Calvo Runge-Kutta-Nystrom(4)*

__Implicit Methods__: Implicit Euler(1), Trapezoid(2),
Midpoint(2)*,
Gauss(4)*, Gauss(6)*, Gauss(8)*, RadauIA(5)

__Problems__: Harmonic Oscillator, Pendulum, Double Harmonic
Oscillator,
Kepler's Problem, Modified Kepler Problem, Henon-Heiles

The Examples problems are from the book *Numerical
Hamiltonian Problems*, J.M. Sanz-Serna and M.P. Calvo, Chapman &
Hall,
1994.