Butcher's simplifying assumption for symplectic integrators

Implicit Runge-Kutta methods with vanishingM matrix are discussed for preserving the symplectic structure of Hamiltonian systems. The number of the order conditions independent of the number of stages can be reduced considerably for the symplectic IRK method through the analysis utilizing the rooted tree and the corresponding elementary differentials. Butcher's simplifying condition further reduces the number of independent order conditions.