Energy conservation by symmetric methods used withsmall step size

The second proof is based on the observation that the methods (2.2), though not themselves symplectic with the exception of the Stormer/Verlet method

(a = 0, f3 = 0), are closely related to symplectic methods. Backward analysis, which interprets the numerical solution of a symplectic one-step method as the

almost-exact solution of a modified Hamiltonian system, then yields the result.