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Jacobi-Lie Hamiltonian systems on real low-dimensional Jacobi-Lie groups and their Lie symmetries

H. Amirzadeh-Fard, Gh. Haghighatdoost, A. Rezaei-Aghdam

Abstract

We study Jacobi-Lie Hamiltonian systems admitting Vessiot-Guldberg Lie algebras of Hamiltonian vector fields related to Jacobi structures on real low-dimensional Jacobi-Lie groups. Also, we find some examples of Jacobi-Lie Hamiltonian systems on real two- and three- dimensional Jacobi-Lie groups. Finally, we present Lie symmetries of Jacobi-Lie Hamiltonian systems on some three-dimensional real Jacobi-Lie groups.

Jacobi-Lie Hamiltonian systems on real low-dimensional Jacobi-Lie groups and their Lie symmetries

Abstract

We study Jacobi-Lie Hamiltonian systems admitting Vessiot-Guldberg Lie algebras of Hamiltonian vector fields related to Jacobi structures on real low-dimensional Jacobi-Lie groups. Also, we find some examples of Jacobi-Lie Hamiltonian systems on real two- and three- dimensional Jacobi-Lie groups. Finally, we present Lie symmetries of Jacobi-Lie Hamiltonian systems on some three-dimensional real Jacobi-Lie groups.

Paper Structure

This paper contains 8 sections, 7 theorems, 53 equations.

Table of Contents

  1. Introduction
  2. Definitions and Notations on Lie and Jacobi-Lie Hamiltonian system
  3. Lie systems and Lie-Hamiltonian systems
  4. Jacobi and Jacobi-Lie structures
  5. Jacobi-Lie Hamiltonian systems
  6. Jacobi-Lie Hamiltonian systems on real low-dimensional Jacobi-Lie groups
  7. Lie symmetry for Jacobi-Lie Hamiltonian systems
  8. Concluding remarks

Key Result

Theorem 2.6

(The Lie-Scheffers Theorem) A system $X$ on $M$ admits a superposition rule if and only if it can be written in the form for a set $b_1(t), . . . , b_r(t)$ of t-dependent functions and a family of vector fields $X_1, . . . , X_r$ on M spanning an r-dimensional real Lie algebra: a VG Lie algebra of X. That is to say, a system X on $M$ possesses a superposition rule if and only if it is a Lie syst

Theorems & Definitions (22)

  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Definition 2.4
  • Definition 2.5
  • Theorem 2.6
  • Definition 2.7
  • Definition 2.8
  • Definition 2.9
  • Definition 2.10
  • ...and 12 more