Having introduced the basic components of classical mechanics that are important in molecular dynamics, we now take up the topic of how to do classical mechanics on a computer. The basic object we will be seeking in this context is a numerical integrator, that is, an algorithm for determining the approximate positions and velocities in a system at any instant in time given the initial conditions. We have already introduced some simple algorithms for integrating a classical system in time. We now seek to formalize the generalize the approach in order to achieve a scheme that is general enough to handle a wide variety of situations, including separation of time scales, non-Hamiltonian dynamics and constrained dynamics.
We begin the discussion by focusing on Hamiltonian systems only. We have already seen that Hamiltonian systems possess certain fundamental properties, including the symplectic property and time reversibility. It is important that any numerical integrator we devise share these properties. The simple algorithm we introduced at the beginning of the class:
the velocity Verlet algorithm, satisfies both properties and is, therefore, a good integrator for Hamiltonian systems.
In the proceeding, we shall be introducing a general formalism for deriving symplectic, reversible integrators starting from an operator-based formulation of classical mechanics. Let us begin by examining a simple one-particle system in one dimension with a Hamiltonian
for which the equattions of motion are
Recall that any function A(x,p) evolves in time according to
where the Poisson bracket is
This evolution must give back Hamilton's equations of motion. In order to verify this, choose A(x,p)=x. Then
since . Thus, as expected. Similarly, by choosing A(x,p)=p,
Define a two-dimensional phase space vector . We can, therefore, write Hamilton's equations in terms of :
Next, define an operator acting on by
The operator L is known as the Liouville operator. The presence of the in the definition will become clear shortly. Abstractly, we may write
which means that we take whatever iL acts on and place in where the ... appear in the Poisson bracket. Note that iL can also be expressed as a differential operator:
Using Hamilton's equations, iL can also be expressed as
Therefore, the equations of motion can be written in operator form as
In this form, we see that the equations of motion can be solved formally to yield
where are the initial conditions. The operator is called the classical propagator, and the presence of the i gives a nice analogy with the quantum mechanical propagator .