Computational Physics

220 Molecular dynamics simulations

x

p

t

Figure 8.3. The area conservation law for a symplectic flow. The integral

∮

pdx

for any loop around the tube representing the flow of a closed loop in thep,x

plane remains constant. This integral represents the area of the projection of the

loop onto thexpplane. Note that the loops do not necessarily lie on a plane of

constant time.

We have seen that symplecticity is a symmetry of Hamiltonian mechanics in
continuum time; now we consider numerical integration methods for Hamiltonian
systems (discrete time). As mentioned above, it is not clear whether full sym-
plecticity is necessary for a reliable description of the dynamics of a system by a
numerical integration. However, it will be clear that the preservation of the sym-
metries present in continuum time mechanics is the most reliable option. The fact
mentioned above, that symplecticity implies conservation of the discrete version
of the total energy, is an additional feature in favour of symplectic integrators for
studying dynamical systems.
It should be noted that symplecticity does not guarantee time reversibility or
vice versa. Time reversibility shows up as the Hamiltonian being invariant when
replacingpby−p, and a Hamiltonian containing odd powers ofpmight still be
symplectic.

*8.4.3 Derivation of symplectic integrators

The first symplectic integrators were found by requiring that an integrator of some
particular form be symplectic. The complexity of the resulting algebraic equations
for the parameters in the integration scheme was found to increase dramatically with