In this work we make us 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.

翻译：本文利用利文斯原理（有时也称为汉密尔顿-庞特里亚金原理）构建了一种适用于机械系统的保结构积分器。与规范汉密尔顿运动方程不同，基于利文斯原理的欧拉-拉格朗日方程避免了质量矩阵求逆过程。这一特性对于处理多体系统建模与仿真中因奇异质量矩阵带来的困难具有关键优势。此外，利文斯原理统一了拉格朗日与汉密尔顿力学视角，且本框架无需建立系统的汉密尔顿函数。该新型算法能在数值层面守恒广义能量函数，并致力于保持对应系统对称性的动量映射。我们进一步将该方法拓展至含完整约束的机械系统，并通过代表性算例验证了新方法的性能。