Sympletic Structure
One important property of a Hamiltonian dynamical system is that it has a sympletic structure. Writing

the evolution equation of the dynamical system can be written as
where
and IN the N×N identity matrix.
One important consequence of this property is that an infinitesimal phase-space volume is preserved. A corollary of this is the Liouville's theorem:
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Liouville's theorem:
Liouville's theorem states that on a Hamiltonian system, the phase-space volume of a closed surface is preserved under time evolution.
where the third equality comes from the divergence theorem. |
Read more about this topic: Hamiltonian System
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