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Variational Collision Integrators for Nonholonomic Lagrangian Systems

Álvaro Rodríguez Abella, Leonardo Colombo

TL;DR

The paper addresses the challenge of designing structure-preserving numerical integrators for nonholonomic Lagrangian systems subject to elastic collisions. It develops a discrete Lagrange-d'Alembert-Pontryagin variational principle using retractions to discretize constraints and introduces discrete collision handling via a collision time ttilde = t_i + alpha h with alpha in (0,1). The resulting theory yields both the (+) and (-) discrete nonholonomic implicit Euler-Lagrange equations with explicit collision conditions, and its effectiveness is demonstrated on a bouncing particle, a bouncing ellipse, and a nonholonomic spherical pendulum inside a cylinder, showing near-constant energy over long times. This framework advances variational integrators for systems with impacts and nonholonomic constraints and sets the stage for extensions to control, reduction, and interconnection in a port-Hamiltonian setting.

Abstract

A discrete theory for implicit nonholonomic Lagrangian systems undergoing elastic collisions is developed. It is based on the discrete Lagrange-d'Alembert-Pontryagin variational principle and the dynamical equations thus obtained are the discrete nonholonomic implicit Euler-Lagrange equations together with the discrete conditions for the elastic impact. To illustrate the theory, variational integrators with collisions are built for several examples, including a bouncing ellipse and a nonholonomic spherical pendulum evolving inside a cylinder.

Variational Collision Integrators for Nonholonomic Lagrangian Systems

TL;DR

The paper addresses the challenge of designing structure-preserving numerical integrators for nonholonomic Lagrangian systems subject to elastic collisions. It develops a discrete Lagrange-d'Alembert-Pontryagin variational principle using retractions to discretize constraints and introduces discrete collision handling via a collision time ttilde = t_i + alpha h with alpha in (0,1). The resulting theory yields both the (+) and (-) discrete nonholonomic implicit Euler-Lagrange equations with explicit collision conditions, and its effectiveness is demonstrated on a bouncing particle, a bouncing ellipse, and a nonholonomic spherical pendulum inside a cylinder, showing near-constant energy over long times. This framework advances variational integrators for systems with impacts and nonholonomic constraints and sets the stage for extensions to control, reduction, and interconnection in a port-Hamiltonian setting.

Abstract

A discrete theory for implicit nonholonomic Lagrangian systems undergoing elastic collisions is developed. It is based on the discrete Lagrange-d'Alembert-Pontryagin variational principle and the dynamical equations thus obtained are the discrete nonholonomic implicit Euler-Lagrange equations together with the discrete conditions for the elastic impact. To illustrate the theory, variational integrators with collisions are built for several examples, including a bouncing ellipse and a nonholonomic spherical pendulum evolving inside a cylinder.

Paper Structure

This paper contains 13 sections, 2 theorems, 36 equations, 4 figures, 1 algorithm.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Retractions and discretization maps
  4. Discrete nonholonomic constraints
  5. Variational formulation of discrete Dirac mechanics
  6. Discrete nonholonomic implicit Lagrangian systems with collisions
  7. Discrete configuration and phase spaces
  8. Discrete nonholonomic implicit Euler--Lagrange equations with collisions
  9. Examples
  10. Bouncing particle
  11. Bouncing 2-dimensional rigid body
  12. Nonholonomic spherical pendulum hitting a cylindrical surface
  13. Conclusions and future work

Key Result

Theorem 2.1

Figures (4)

  • Figure 1: Trajectory of an ellipse rotating and colliding with the floor. The initial and final times are $t_0=0$ and $t_N=2$.
  • Figure 2: Long-time evolution of the energy of the ellipse colliding with the floor. The initial and final times are $t_0=0$ and $t_N=25$. The system underwent 17 impacts during that period of time.
  • Figure 3: Evolution of the angles of a nonholonomic spherical pendulum colliding with a cylindrical surface. The initial and final times are $t_0=0$ and $t_N=5$, and the time-step is $h=10^{-3}$.
  • Figure 4: Long-time evolution of the energy of a nonholonomic spherical pendulum colliding with a cylindrical surface. The initial and final times are $t_0=0$ and $t_N=100$, and the time-step is $h=10^{-4}$. The system underwent 76 impacts during that period of time.

Theorems & Definitions (4)

  • Theorem 2.1
  • Definition 3.1
  • Theorem 3.1
  • proof