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
].
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Numerical Approaches to Spacetime Singularities
Beverly K. Berger http://www.livingreviews.org/lrr-1998-7 © Max-Planck-Gesellschaft. ISSN 1433-8351 Problems/Comments to livrev@aei-potsdam.mpg.de |