Two types of variational integrators and their equivalence

TL;DR

This paper compares two types of variational integrators, showing their equivalence and highlighting their symplectic, symmetric properties, super-convergence, and relation to continuous-stage Runge-Kutta methods for classical mechanics.

Contribution

It introduces and proves the equivalence of two variational integrator types, linking them to advanced numerical methods for mechanical systems.

Findings

The two variational integrators are equivalent for classical mechanical systems.

They are symplectic and symmetric, ensuring good long-term energy behavior.

They achieve super-convergence order 2s, depending on polynomial degree.

Abstract

In this paper, we introduce two types of variational integrators, one originating from the discrete Hamilton's principle while the other from Galerkin variational approach. It turns out that these variational integrators are equivalent to each other when they are used for integrating the classical mechanical system with Lagrangian function $L(q,\dot{q})=\frac{1}{2}\dot{q}^TM\dot{q}-U(q)$ ($M$ is an invertible symmetric constant matrix). They are symplectic, symmetric, possess super-convergence order $2s$ (which depends on the degree of the approximation polynomials), and can be related to continuous-stage partitioned Runge-Kutta methods.