Energy-consistent integration of mechanical systems based on Livens principle

TL;DR

This paper introduces a new structure-preserving integrator for mechanical systems based on Livens principle, which avoids mass matrix inversion and unifies Lagrangian and Hamiltonian mechanics, ensuring energy and momentum conservation.

Contribution

The work develops a novel integrator using Livens principle that handles singular mass matrices and extends to constrained systems, improving simulation robustness.

Findings

The integrator conserves a general energy function.

It effectively handles systems with singular mass matrices.

Performance demonstrated on representative mechanical examples.

Abstract

In this work we make use of Livens principle (sometimes also referred to as Hamilton-Pontryagin principle) in order to obtain a novel structure-preserving integrator for mechanical systems. In contrast to the canonical Hamiltonian equations of motion, the Euler-Lagrange equations pertaining to Livens principle circumvent the need to invert the mass matrix. This is an essential advantage with respect to singular mass matrices, which can yield severe difficulties for the modelling and simulation of multibody systems. Moreover, Livens principle unifies both Lagrangian and Hamiltonian viewpoints on mechanics. Additionally, the present framework avoids the need to set up the system's Hamiltonian. The novel scheme algorithmically conserves a general energy function and aims at the preservation of momentum maps corresponding to symmetries of the system. We present an extension to mechanical systems subject to holonomic constraints. The performance of the newly devised method is studied in representative examples.