LAUSR

A new family of high order one-step symplectic integration schemes for separable Hamiltonian systems with Hamiltonians of the form $$T(p) + U(q)$$ T ( p ) + U ( q ) is presented. The new integration methods are defined in terms of an explicitly defined generating function (of the third kind). They are implicit in q (but explicit in p and the internal states), and require the evaluation of the gradients of T ( p ) and U ( q ) and the actions of their Hessians on vectors (the later being relatively cheap in the case of many-body problems). A time-symmetric symplectic method is constructed that has order 10 when applied to Hamiltonian systems with quadratic kinetic energy T ( p ). It is shown by numerical experiments that the new methods have the expected order of convergence.