For a field
*q*
(*t*) and its conjugate momentum
*p*
(*t*) split the Hamiltonian operator into kinetic and potential
energy subhamiltonians. Thus,

If the vector
*X*
= (*p*,
*q*) defines the variables at time
*t*, then the time evolution is given by

where is the Poisson bracket. The usual exponentiation yields an evolution operator

for
the generator of the time evolution. Higher order accuracy may
be obtained by a better approximation to the evolution operator [172,
173]. This method is useful when exact solutions for the
subhamiltonians are known. For the given
*H*, variation of
yields the solution

while that of yields

Note that
is exactly solvable for any potential
*V*
no matter how complicated, although the required differenced
form of the potential gradient may be non-trivial. One evolves
from
*t*
to
using the exact solutions to the subhamiltonians according to
the prescription given by the approximate evolution operator (8). Extension to more degrees of freedom and to fields is
straightforward [32,
22].

Numerical Approaches to Spacetime Singularities
Beverly K. Berger
http://www.livingreviews.org/lrr-1998-7
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