New high order symplectic integrators via generating functions with its application in many-body problem

Abstract

A new family of high order one-step symplectic integration schemes for separable Hamiltonian systems with Hamiltonians of the form \(T(p) + U(q)\) 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.