Symplectic Runge–Kutta Schemes and Variational Integrations of Pendulum Equation

Abstract

To preserve the symplecticity property, it is natural to require numerical integration of Hamiltonian systems should be symplectic. The most important classes of symplectic methods are symplectic Runge–Kutta methods and variational integrations. Symplectic Euler method and midpoint rule are the most widely used symplectic Runge–Kutta methods, while the most widely used variational integrations are trapezoidal integration and midpoint-type variational integration. For pendulum equation, which is a model of Hamiltonian systems, we propose symplectic Euler method, midpoint scheme, trapezoidal integration, and midpoint-type variational integration of the model in this paper. At last, it will be proved that symplectic Euler method is equivalent to trapezoidal integration, while midpoint scheme is equivalent to midpoint-type variational integration.KeywordsHamiltonian systemsSymplecticityRunge–Kutta methodVariational integration