Area Preserving Flows

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 regionin phase space, letbe the area ofand letbe the region of phase space obtained by taking a regionat timeand letting it evolve through a timecontains pointssuch thatinandwhereIf the system is conservative thenwhereis the flow of the system.