Operators and numerical propagation methods next up previous
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Operators and numerical propagation methods

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:

eqnarray34

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

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for which the equattions of motion are

displaymath41

Recall that any function A(x,p) evolves in time according to

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where the Poisson bracket is

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This evolution must give back Hamilton's equations of motion. In order to verify this, choose A(x,p)=x. Then

eqnarray56

since tex2html_wrap_inline212 . Thus, tex2html_wrap_inline214 as expected. Similarly, by choosing A(x,p)=p,

eqnarray66

Thus, tex2html_wrap_inline218 .

Define a two-dimensional phase space vector tex2html_wrap_inline220 . We can, therefore, write Hamilton's equations in terms of tex2html_wrap_inline222 :

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Next, define an operator acting on tex2html_wrap_inline222 by

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The operator L is known as the Liouville operator. The presence of the tex2html_wrap_inline228 in the definition will become clear shortly. Abstractly, we may write

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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:

eqnarray84

Using Hamilton's equations, iL can also be expressed as

eqnarray97

Therefore, the equations of motion can be written in operator form as

eqnarray105

In this form, we see that the equations of motion can be solved formally to yield

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where tex2html_wrap_inline236 are the initial conditions. The operator tex2html_wrap_inline238 is called the classical propagator, and the presence of the i gives a nice analogy with the quantum mechanical propagator tex2html_wrap_inline242 .


next up previous
Next: Properties of the Liouville Up: No Title Previous: No Title

Mark Tuckerman
Thu Oct 17 00:09:53 EDT 2002