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Energy conserving nonholonomic integrators

Jorge Cortes

TL;DR

The paper addresses energy conservation in numerical integration of nonholonomic Lagrangian systems by introducing an extended discrete Lagrange-d'Alembert principle for nonautonomous dynamics. It develops extended variational integrators based on time-dependent discrete mechanics, yielding the extended discrete Euler–Lagrange map and EDLA equations that preserve energy in autonomous cases while maintaining discrete analogues of the symplectic form and momentum. The analysis introduces a discrete nonholonomic momentum map and a reduction framework (REDLA) for generalized Chaplygin systems under discrete assumptions, linking reduced dynamics to gyroscopic forcing and highlighting energy preservation in autonomous settings. Overall, the work provides a structure-preserving, energy-conserving approach for nonholonomic integration and motivates future numerical error analyses via backward error techniques.

Abstract

We address the problem of constructing numerical integrators for nonholonomic Lagrangian systems that enjoy appropriate discrete versions of the geometric properties of the continuous flow, including the preservation of energy. Building on previous work on time-dependent discrete mechanics, our approach is based on a discrete version of the Lagrange-d'Alembert principle for nonautonomous systems.

Energy conserving nonholonomic integrators

TL;DR

The paper addresses energy conservation in numerical integration of nonholonomic Lagrangian systems by introducing an extended discrete Lagrange-d'Alembert principle for nonautonomous dynamics. It develops extended variational integrators based on time-dependent discrete mechanics, yielding the extended discrete Euler–Lagrange map and EDLA equations that preserve energy in autonomous cases while maintaining discrete analogues of the symplectic form and momentum. The analysis introduces a discrete nonholonomic momentum map and a reduction framework (REDLA) for generalized Chaplygin systems under discrete assumptions, linking reduced dynamics to gyroscopic forcing and highlighting energy preservation in autonomous settings. Overall, the work provides a structure-preserving, energy-conserving approach for nonholonomic integration and motivates future numerical error analyses via backward error techniques.

Abstract

We address the problem of constructing numerical integrators for nonholonomic Lagrangian systems that enjoy appropriate discrete versions of the geometric properties of the continuous flow, including the preservation of energy. Building on previous work on time-dependent discrete mechanics, our approach is based on a discrete version of the Lagrange-d'Alembert principle for nonautonomous systems.

Paper Structure

This paper contains 6 sections, 7 theorems, 33 equations.

Table of Contents

  1. Introduction
  2. Mechanical integrators
  3. Variational integrators
  4. A discrete Lagrange-d'Alembert principle for nonautonomous systems
  5. Geometric properties
  6. Conclusions

Key Result

Theorem 1

Consider a Hamiltonian system with a symmetry group $G$ such that the dynamics $X_H$ is nonintegrable on the reduced space (in the sense that any other conserved quantity is functionally dependent on $H$). Assume that a numerical integrator for this system is energy-symplectic-momentum preserving an

Theorems & Definitions (13)

  • Theorem 1: GeMa
  • Theorem 2: MaWe
  • Definition 1
  • Remark 1
  • Remark 2: Well-posedness of the discrete problem
  • Proposition 1
  • proof
  • Corollary 1
  • proof
  • Proposition 2
  • ...and 3 more