On the Importance of Being Symplectic

Symplecticness is a strong condition and we have seen, in the construction of the perturbed Hamiltonian the last section, that it implies a constraint on the growth of errors in energy. This does not, however, automatically mean that there are no other suitable principles that could underpin the design of good numerical methods for molecular dynamics. The flow of a Hamiltonian system of the form H = p M p/2 + U(q) will preserve all of the following  

The volume in phase space (as the vector field is divergence free). 

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