For a field q (t) and its conjugate momentum p (t), the Hamiltonian operator splits into kinetic and potential energy sub-Hamiltonians. Thus, for an arbitrary potential V (q),
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
being the generator of the time evolution. Higher order accuracy
may be obtained by a better approximation to the evolution
operator [234,
235]. This method is useful when exact solutions for the
sub-Hamiltonians 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 sub-Hamiltonians according to
the prescription given by the approximate evolution
operator (8
). Extension to more degrees of freedom and to fields is
straightforward [42
,
30
].
As pointed out in [42,
35
,
41
,
43
], spiky features in collapsing inhomogeneous cosmologies will
cause any fixed spatial resolution to become inadequate. Such
evolutions are therefore candidates for adaptive mesh refinement
such as that implemented by Hern and Stuart [143
,
142].
![]() |
Numerical Approaches to Spacetime Singularities
Beverly K. Berger http://www.livingreviews.org/lrr-2002-1 © Max-Planck-Gesellschaft. ISSN 1433-8351 Problems/Comments to livrev@aei-potsdam.mpg.de |