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Nonsmooth Herglotz variational principle

Asier López-Gordón, Leonardo Colombo, Manuel de León

TL;DR

The paper extends the Herglotz variational principle, which models dissipative dynamics through action-dependent Lagrangians, to non-smooth trajectories arising from impacts by adopting a Fetecau–Marsden–Ortiz–West framework that fixes impacts in a parameter, creating a smooth path space. It derives the nonsmooth Herglotz equations and corresponding nonsmooth Hamiltonian equations via a Legendre transform, including precise jump conditions at the impact surface to conserve momentum-like flux and account for energy dissipation. A case study on billiards with dissipation demonstrates the method, deriving explicit impact rules and dissipated quantities (e.g., angular momentum-like integrals) and showing the compatibility with a hybrid-contact viewpoint. The work links action-dependent Lagrangian dynamics with contact geometry and hybrid systems, enabling variational treatment of dissipative, impact-driven mechanics and suggesting avenues for structure-preserving integrators and energy accounting in future research.

Abstract

In this paper, the theory of smooth action-dependent Lagrangian mechanics (also known as contact Lagrangians) is extended to a non-smooth context appropriate for collision problems. In particular, we develop a Herglotz variational principle for non-smooth action-dependent Lagrangians which leads to the preservation of energy and momentum at impacts. By defining appropriately a Legendre transform, we can obtain the Hamilton equations of motion for the corresponding non-smooth Hamiltonian system. We apply the result to a billiard problem in the presence of dissipation.

Nonsmooth Herglotz variational principle

TL;DR

The paper extends the Herglotz variational principle, which models dissipative dynamics through action-dependent Lagrangians, to non-smooth trajectories arising from impacts by adopting a Fetecau–Marsden–Ortiz–West framework that fixes impacts in a parameter, creating a smooth path space. It derives the nonsmooth Herglotz equations and corresponding nonsmooth Hamiltonian equations via a Legendre transform, including precise jump conditions at the impact surface to conserve momentum-like flux and account for energy dissipation. A case study on billiards with dissipation demonstrates the method, deriving explicit impact rules and dissipated quantities (e.g., angular momentum-like integrals) and showing the compatibility with a hybrid-contact viewpoint. The work links action-dependent Lagrangian dynamics with contact geometry and hybrid systems, enabling variational treatment of dissipative, impact-driven mechanics and suggesting avenues for structure-preserving integrators and energy accounting in future research.

Abstract

In this paper, the theory of smooth action-dependent Lagrangian mechanics (also known as contact Lagrangians) is extended to a non-smooth context appropriate for collision problems. In particular, we develop a Herglotz variational principle for non-smooth action-dependent Lagrangians which leads to the preservation of energy and momentum at impacts. By defining appropriately a Legendre transform, we can obtain the Hamilton equations of motion for the corresponding non-smooth Hamiltonian system. We apply the result to a billiard problem in the presence of dissipation.
Paper Structure (10 sections, 3 theorems, 57 equations, 2 figures)

This paper contains 10 sections, 3 theorems, 57 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Herglotz principle for nonsmooth action-dependent Lagrangians
  3. Herglotz variational principle for smooth Lagrangians
  4. Herglotz principle for nonsmooth Lagrangians
  5. Nonsmooth Hamiltonian equations for contact Hamiltonian systems
  6. Case Study: Billard with dissipation
  7. Elliptical billiard
  8. Relation with contact Lagrangian and Hamiltonian systems
  9. Hybrid contact systems
  10. Conclusions and Future work

Key Result

Theorem 1

Let $L:TQ \times \mathbb{R}$ be a smooth and regular Lagrangian function. Let $c=(c_q, c_t)$ be a curve in $\widehat{\Omega} \left(q_1, q_2, [0,1] \right)$, and let $\chi (\tau)= \left(c_q(\tau), \frac{c_q'(\tau)}{c_t'(\tau)}, \widehat{\mathcal{Z}}(c)(\tau)\right) \subset TQ\times \mathbb{R}$. Then for $\tau \in [0, \tau_i) \cup (\tau_i,1]$, and where ${\chi(\tau_i^\pm) = \lim_{\tau \to \tau_i^\

Figures (2)

  • Figure 1: Numerical simulation for the trajectory of a particle in the billiard, with $\gamma = 10^{-4}$.
  • Figure 2: Numerical simulation for the trajectory of a particle in an elliptical billiard, with $\gamma = 10^{-4}$.

Theorems & Definitions (11)

  • Theorem 1: Nonsmooth Herglotz variational principle
  • proof
  • Remark 1
  • Definition 1
  • Proposition 2: Hamiltonian nonsmooth Herglotz principle
  • Remark 2
  • Theorem 3
  • Definition 2
  • Remark 3
  • Definition 3
  • ...and 1 more