Computational Physics

8.4 Integration methods: symplectic integrators 217

method in which a time-dependent friction parameter occurs, obeying a first order
differential equation:

̈x(t)=−γ(t)x ̇(t)+F[x(t)] (8.42a)

γ( ̇t)=g[ ̇x(t)]. (8.42b)

The solution can conveniently be presented in the leap-frog formulation. As the
momentum is given at half-integer time steps in this formulation, we can solve for
γin the following way:

γ(h)=γ( 0 )+hg[p(h/ 2 )]+O(h^2 ), (8.43)

and this is to be combined withEqs. (8.41). Velocity-Verlet formulations(Eqs. (8.9))
for equations of motions including friction terms can be found straightforwardly.
Thisisleftasanexercisetothereader–seealsoRef.[26].

*8.4.2 Symplectic geometry; symplectic integrators

In recent years, major improvement has been achieved in understanding the merits
of the various methods for integrating equations of motion which can be derived
from a Hamiltonian. This development started in the early 1980s with the observa-
tions made independently by Ruth[27]and Feng[28]that methods for solving
Hamiltonian equations of motion should preserve the geometrical structure of
the continuum solution in phase space. This geometry is the so-calledsymplectic
geometry. Below we shall explain what this geometry is about, and what the prop-
erties of symplectic integrators are. InSection 8.4.3we shall see how symplectic
integrators can be constructed. We restrict ourselves again to a two-dimensional
phase space (one particle moving in one dimension) spanned by the coordinates
pandx, but it should be realised that the analysis is trivially generalised to arbit-
rary numbers of particles in higher dimensional space with phase space points
(p 1 ,...,pm,r 1 ,...,rm).^4 The equations of motion for the particle are derived from a
Hamiltonian which for a particle moving in a potential (in the absence of constraints)
reads

H(p,x)=

p^2

2

+V(x). (8.44)

The Hamilton equations of motion are then given as

p ̇=−

∂H(p,x)

∂x

(8.45a)

̇x=

∂H(p,x)

∂p

(8.45b)

(^4) Although we use the notationrifor the coordinates, they may be generalised coordinates.