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Lie integrability by quadratures for symplectic, cosymplectic, contact and cocontact Hamiltonian systems

R. Azuaje

Abstract

In this paper we present the theorem on Lie integrability by quadratures for time-independent Hamiltonian systems on symplectic and contact manifolds, and for time-dependent Hamiltonian systems on cosymplectic and cocontact manifolds. We show that having a solvable Lie algebra of constants of motion for a Hamiltonian system is equivalent to having a solvable Lie algebra of symmetries of the vector field defining the dynamics of the system, which allows us to find the solutions of the equations of motion by quadratures.

Lie integrability by quadratures for symplectic, cosymplectic, contact and cocontact Hamiltonian systems

Abstract

In this paper we present the theorem on Lie integrability by quadratures for time-independent Hamiltonian systems on symplectic and contact manifolds, and for time-dependent Hamiltonian systems on cosymplectic and cocontact manifolds. We show that having a solvable Lie algebra of constants of motion for a Hamiltonian system is equivalent to having a solvable Lie algebra of symmetries of the vector field defining the dynamics of the system, which allows us to find the solutions of the equations of motion by quadratures.
Paper Structure (6 sections, 7 theorems, 67 equations)

This paper contains 6 sections, 7 theorems, 67 equations.

Table of Contents

  1. Introduction
  2. Symplectic, cosymplectic, contact and cocontact Hamiltonian systems
  3. Lie algebras of symmetries for vector fields in $\mathbb{R}^{n}$
  4. Lie algebras of symmetries for symplectic and cosymplectic Hamiltonian systems
  5. Lie algebras of symmetries for contact and cocontact Hamiltonian systems
  6. Conclusions

Key Result

Theorem 1

Let $u_{1},\ldots u_{n}$ be linearly independent smooth vector fields on $\mathbb{R}^{n}$. If $u_{1},\ldots,u_{n}$ are symmetries of the vector field $v$ and they generate a solvable Lie algebra with the Lie bracket $[,]$ of vector fields, then the system defined by $\dot{x}=v(x)$ is integrable by q

Theorems & Definitions (15)

  • Definition 1
  • Definition 2
  • Definition 3
  • Definition 4
  • Definition 5
  • Definition 6
  • Theorem 1
  • Theorem 2
  • Theorem 3
  • Lemma 1
  • ...and 5 more