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Hamilton-Jacobi Formalism on Locally Conformally Symplectic Manifolds

Oğul Esen, Manuel de León, Cristina Sardón, Marcin Zajac

Abstract

In this article we provide a Hamilton-Jacobi formalism in locally conformally symplectic manifolds. Our interest in the Hamilton-Jacobi theory comes from the suitability of this theory as an integration method for dynamical systems, whilst our interest in the locally conformal character will account for physical theories described by Hamiltonians defined on well-behaved line bundles, whose dynamic takes place in open subsets of the general manifold. We present a local l.c.s. Hamilton-Jacobi in subsets of the general manifold, and then provide a global view by using the Lichnerowicz-deRham differential. We show a comparison between the global and local description of a l.c.s. Hamilton--Jacobi theory, and how actually the local behavior can be glued to retrieve the global behavior of the Hamilton-Jacobi theory.

Hamilton-Jacobi Formalism on Locally Conformally Symplectic Manifolds

Abstract

In this article we provide a Hamilton-Jacobi formalism in locally conformally symplectic manifolds. Our interest in the Hamilton-Jacobi theory comes from the suitability of this theory as an integration method for dynamical systems, whilst our interest in the locally conformal character will account for physical theories described by Hamiltonians defined on well-behaved line bundles, whose dynamic takes place in open subsets of the general manifold. We present a local l.c.s. Hamilton-Jacobi in subsets of the general manifold, and then provide a global view by using the Lichnerowicz-deRham differential. We show a comparison between the global and local description of a l.c.s. Hamilton--Jacobi theory, and how actually the local behavior can be glued to retrieve the global behavior of the Hamilton-Jacobi theory.

Paper Structure

This paper contains 14 sections, 5 theorems, 67 equations.

Table of Contents

  1. Introduction
  2. Geometry of locally conformally symplectic manifolds
  3. Basics of locally conformally symplectic manifolds
  4. Locally conformally symplectic structures on cotangent bundles.
  5. Dynamics on locally conformally symplectic manifolds
  6. Jacobi structures and l.c.s. manifolds
  7. Hamilton-Jacobi theory on l.c.s. manifolds
  8. The local picture of a l.c.s. HJ theory
  9. A global picture for a l.c.s. Hamilton-Jacobi theory
  10. Comparison of local and global pictures
  11. An example
  12. Complete solutions
  13. Discussion, and Future Works
  14. Acknowledgments

Key Result

Theorem \oldthetheorem

For a closed one-form $\gamma=dW$ on $Q$ the following conditions are equivalent:

Theorems & Definitions (8)

  • Theorem \oldthetheorem
  • Lemma \oldthetheorem
  • proof
  • Theorem \oldthetheorem
  • Theorem \oldthetheorem
  • proof
  • Theorem \oldthetheorem
  • proof