🤖 AI Summary
To address modeling difficulties arising from singular mass matrices in mechanical systems and the conceptual disjunction between Lagrangian and Hamiltonian frameworks, this paper proposes a novel structure-preserving numerical integrator grounded in the Livens (Hamilton–Pontryagin) variational principle. The method avoids explicit Hamiltonian construction and mass matrix inversion, naturally accommodates holonomic constraints, and unifies Lagrangian and Hamiltonian descriptions. Its core contribution lies in the first systematic application of the Livens principle to design integrators that are energy-consistent and rigorously preserve symmetry-induced momentum maps. Theoretical analysis guarantees long-term conservation of a generalized energy function and the associated momentum map. Numerical experiments demonstrate high accuracy, strong robustness, and excellent long-term stability for singular multibody systems, even under challenging configurations where conventional methods fail.
📝 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.