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The evolution operator connecting the Lagrangian and Hamiltonian formalisms for contact systems

Xavier Gràcia, Ángel Martínez-Muñoz, Xavier Rivas, Narciso Román-Roy

TL;DR

This work extends the Lagrangian–Hamiltonian evolution operator $K$ to contact (dissipative) mechanics, providing a geometric framework for Herglotz-type dynamics and a bridge between Lagrangian and Hamiltonian constraint formalisms. It defines $K$ as a vector field along the Legendre map that satisfies a second-order condition and dynamical equations, recovering the Euler–Lagrange and Hamilton equations in regular cases and enabling constraint propagation between formalisms. The authors develop intrinsic, Reeb-independent formulations of the contact Hamiltonian equations, extend almost-regular Lagrangian theory to the contact setting, and prove that Hamiltonian constraints yield Lagrangian constraints via $K$, including conditions for $\mathcal{F}L$-projectability. Two detailed examples—the damped simple pendulum and Cawley’s Lagrangian with dissipation—illustrate the constraint algorithms and the action of $K$ on Hamiltonian constraints to produce corresponding Lagrangian constraints. The results deepen the understanding of singular contact dynamics and pave the way for precontact and pre-Lagrangian extensions of constraint algorithms in dissipative systems.

Abstract

Some mechanical systems with dissipation can be described within the framework of the so-called contact mechanics: a modified form of the Euler-Lagrange equations stemming from Herglotz's variational principle, which admits a geometric formulation in terms of contact geometry. On the other hand, the study of singular Lagrangian systems and Dirac's theory of constraints can be enhanced by using the evolution operator $K$ that connects the Lagrangian and Hamiltonian formalisms. The main purpose of this paper is to transpose this evolution operator to the case of contact mechanics, and to study some of its main properties. In particular, we show that it provides a geometric description of the evolution equations and it relates the Hamiltonian and Lagrangian constraints. To illustrate the theory, we provide examples of singular contact systems based on modified versions of the simple pendulum and the Cawley Lagrangian.

The evolution operator connecting the Lagrangian and Hamiltonian formalisms for contact systems

TL;DR

This work extends the Lagrangian–Hamiltonian evolution operator to contact (dissipative) mechanics, providing a geometric framework for Herglotz-type dynamics and a bridge between Lagrangian and Hamiltonian constraint formalisms. It defines as a vector field along the Legendre map that satisfies a second-order condition and dynamical equations, recovering the Euler–Lagrange and Hamilton equations in regular cases and enabling constraint propagation between formalisms. The authors develop intrinsic, Reeb-independent formulations of the contact Hamiltonian equations, extend almost-regular Lagrangian theory to the contact setting, and prove that Hamiltonian constraints yield Lagrangian constraints via , including conditions for -projectability. Two detailed examples—the damped simple pendulum and Cawley’s Lagrangian with dissipation—illustrate the constraint algorithms and the action of on Hamiltonian constraints to produce corresponding Lagrangian constraints. The results deepen the understanding of singular contact dynamics and pave the way for precontact and pre-Lagrangian extensions of constraint algorithms in dissipative systems.

Abstract

Some mechanical systems with dissipation can be described within the framework of the so-called contact mechanics: a modified form of the Euler-Lagrange equations stemming from Herglotz's variational principle, which admits a geometric formulation in terms of contact geometry. On the other hand, the study of singular Lagrangian systems and Dirac's theory of constraints can be enhanced by using the evolution operator that connects the Lagrangian and Hamiltonian formalisms. The main purpose of this paper is to transpose this evolution operator to the case of contact mechanics, and to study some of its main properties. In particular, we show that it provides a geometric description of the evolution equations and it relates the Hamiltonian and Lagrangian constraints. To illustrate the theory, we provide examples of singular contact systems based on modified versions of the simple pendulum and the Cawley Lagrangian.

Paper Structure

This paper contains 15 sections, 15 theorems, 134 equations.

Table of Contents

  1. Introduction
  2. The evolution operator of Lagrangian/Hamiltonian mechanics
  3. Contact dynamics
  4. Hamiltonian systems
  5. Lagrangian systems
  6. Hamiltonian formalism for contact Lagrangian systems
  7. Almost-regular Lagrangian functions
  8. The evolution operator in contact mechanics
  9. Construction of the evolution operator
  10. The evolution operator and the equations of motion
  11. Relating the constraint algorithms
  12. Examples
  13. The simple pendulum with damping
  14. Cawley's Lagrangian with dissipation
  15. Conclusions and outlook

Key Result

Theorem 3.1

Given a contact manifold $(M,\eta)$ with $\dim M=2n+1$, around every point $x\in M$ there exist local coordinates $(q^i, p_i, s)$, with $i = 1,\dotsc,n$, called Darboux coordinates, such that In these coordinates, $R = \partial /\partial s$.

Theorems & Definitions (27)

  • Theorem 3.1: Darboux theorem AM_78LM_87
  • Example 3.2: Canonical contact manifold
  • Proposition 3.3
  • proof
  • Remark 3.4
  • Definition 3.5
  • Definition 3.6
  • Proposition 3.7
  • Definition 3.8
  • Remark 3.9
  • ...and 17 more