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

TL;DR

This paper establishes conditions under which Hamiltonian systems on various geometric manifolds are integrable by quadratures, linking solvable Lie algebras of constants of motion to solvable symmetry Lie algebras.

Contribution

It extends Lie integrability by quadratures to symplectic, cosymplectic, contact, and cocontact Hamiltonian systems, providing a unified geometric framework.

Findings

Solvable Lie algebra of constants of motion implies solvable symmetry algebra.

Solutions of equations of motion can be obtained by quadratures.

Applicability to time-independent and time-dependent 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.