Hamiltonian systems can be considered as those systems whose evolution or flow preserves area in phase space. Any system which satisfies Hamilton's equations preserves area in this way. An explicit statement is given below.
Consider any Hamiltonian system, not necessarily conservative. For any bounded region
in phase space, let
be the area ofand let
be the region of phase space obtained by taking a region
at time
and letting it evolve through a time
contains points
such that
inand
where
If the system is conservative then
where
is the flow of the system.
