Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Liouville-Arnold theorem for homogeneous symplectic and contact Hamiltonian systems

Leonardo Colombo, Manuel de León, Manuel Lainz, Asier López-Gordón

TL;DR

The paper extends Liouville–Arnold theory to completely integrable contact systems by constructing a foliation of the contact manifold into $(n{+}1)$-dimensional Abelian groups via preimages of rays of the involutive integrals and deriving action–angle coordinates after symplectization. A foundational LA theorem for homogeneous integrals on exact symplectic manifolds underpins the construction, which is then transported to the contact setting through symplectization and projection. The authors show that, under completeness, the ray fibers are coisotropic and invariant, yielding linearized dynamics in canonical coordinates and, in favorable cases, Darboux-type forms for the rescaled contact form. The work also provides alternative characterizations via coisotropic submanifolds, reviews related definitions in the literature, and illustrates the theory with a concrete example. Overall, the results offer a robust framework for integrating a broad class of dissipative and non-Reeb-invariant contact systems through action–angle coordinates grounded in homogeneous symplectic geometry.

Abstract

A Hamiltonian system is completely integrable (in the sense of Liouville) if there exist as many independent integrals of motion in involution as the dimension of the configuration space. Under certain regularity conditions, Liouville-Arnold theorem states that the invariant geometric structure associated with Liouville integrability is a fibration by Lagrangian tori (or, more generally, Abelian Lie groups), on which the motion is linear. In this paper, a Liouville-Arnold theorem for contact Hamiltonian systems is proven. More specifically, it is shown that, given a $(2n+1)$-dimensional completely integrable contact system, one can construct a foliation by $(n+1)$-dimensional Abelian Lie groups and induce action-angle coordinates in which the equations of motion are linearized. One important novelty with respect to previous attempts is that the foliation consists of $(n+1)$-dimensional coisotropic submanifolds given by the preimages of rays by the functions in involution. In order to prove the theorem, we first develop a version of Liouville-Arnold theorem for homogeneous functions on exact symplectic manifolds (which is of independent interest), and then apply the symplectization to obtain the contact case.

Liouville-Arnold theorem for homogeneous symplectic and contact Hamiltonian systems

TL;DR

The paper extends Liouville–Arnold theory to completely integrable contact systems by constructing a foliation of the contact manifold into -dimensional Abelian groups via preimages of rays of the involutive integrals and deriving action–angle coordinates after symplectization. A foundational LA theorem for homogeneous integrals on exact symplectic manifolds underpins the construction, which is then transported to the contact setting through symplectization and projection. The authors show that, under completeness, the ray fibers are coisotropic and invariant, yielding linearized dynamics in canonical coordinates and, in favorable cases, Darboux-type forms for the rescaled contact form. The work also provides alternative characterizations via coisotropic submanifolds, reviews related definitions in the literature, and illustrates the theory with a concrete example. Overall, the results offer a robust framework for integrating a broad class of dissipative and non-Reeb-invariant contact systems through action–angle coordinates grounded in homogeneous symplectic geometry.

Abstract

A Hamiltonian system is completely integrable (in the sense of Liouville) if there exist as many independent integrals of motion in involution as the dimension of the configuration space. Under certain regularity conditions, Liouville-Arnold theorem states that the invariant geometric structure associated with Liouville integrability is a fibration by Lagrangian tori (or, more generally, Abelian Lie groups), on which the motion is linear. In this paper, a Liouville-Arnold theorem for contact Hamiltonian systems is proven. More specifically, it is shown that, given a -dimensional completely integrable contact system, one can construct a foliation by -dimensional Abelian Lie groups and induce action-angle coordinates in which the equations of motion are linearized. One important novelty with respect to previous attempts is that the foliation consists of -dimensional coisotropic submanifolds given by the preimages of rays by the functions in involution. In order to prove the theorem, we first develop a version of Liouville-Arnold theorem for homogeneous functions on exact symplectic manifolds (which is of independent interest), and then apply the symplectization to obtain the contact case.
Paper Structure (12 sections, 17 theorems, 116 equations)

This paper contains 12 sections, 17 theorems, 116 equations.

Table of Contents

  1. Introduction
  2. Jacobi and contact manifolds
  3. Jacobi manifolds
  4. Contact manifolds
  5. Contact Hamiltonian systems
  6. Liouville--Arnold theorem for exact symplectic manifolds
  7. Liouville--Arnold theorem for contact Hamiltonian systems
  8. Symplectization of contact manifolds
  9. Proof of the Liouville--Arnold theorem for contact Hamiltonian systems
  10. Integrable systems, Legendrian and coisotropic submanifolds
  11. Other definitions of contact integrable systems
  12. Example

Key Result

Proposition 1

Given a contact manifold $(M,\eta)$, the map $f \mapsto X_f$ is a Lie algebra anti-isomorphism between the set of smooth functions with the Jacobi bracket and the set of infinitesimal conformal contactomorphisms with the Lie bracket, namely, for any $f,g \in \mathscr{C}^\infty(M)$. Its inverse is given by $X \mapsto - \eta(X)$. Furthermore, $X_f$ is an infinitesimal strict contactomorphism if and

Theorems & Definitions (43)

  • Definition 1
  • Definition 2
  • Definition 3
  • Definition 4
  • Proposition 1
  • proof
  • Definition 5
  • Definition 6
  • Definition 7
  • Proposition 2
  • ...and 33 more