Equivariant symplectic difference schemes and generating functions (Q1176810)

Let \(H\) be a Hamiltonian function on a symplectic manifold and \(z(t)\) be the solution of the Hamiltonian system \(dz/dt=J^{-1}H(z)\). One wants to find a symplectic difference scheme \(z\hookrightarrow D_ H(z)\) approximating the solution of the Hamiltonian system. In general this mapping depends on the system of coordinates we use, hence is not covariant. Let \(T\) be a symplectic coordinate transformation, then one says that \(D_ H\) is invariant under \(T\) if \(T^{-1}D_ HT=D_{H\circ T}\). Let us say that a smooth function \(f\) is a first integral if the Poisson bracket \(\{f,H\}\) is null. The essential result used in this paper (and proved elsewhere) states that a first integral \(f\) is preserved up to a constant by \(D_ H\) (i.e. \(f\circ D_ H=f+c\)) if and only if \(D_ H\) is invariant under the phase flow of \(f\). It follows that the mid-point difference scheme preserves quadratic first integrals and moreover that this result can be applied to difference schemes constructed via generating functions (for example in the case of Lie-Poisson Hamiltonian structures).