On particular integrability for (co)symplectic and (co)contact Hamiltonian systems

TL;DR

This paper extends the concepts of particular integrability to cosymplectic, contact, and cocontact Hamiltonian systems, enabling analysis of nonintegrable time-dependent systems and dissipative dynamics through reduced equations.

Contribution

It introduces the notion of particular integrals in these geometries and demonstrates their use in deriving integral curves from lower-dimensional Hamilton equations.

Findings

Existence of particular integrals allows reduction to lower-dimensional systems.

Particular integrability enables solving trajectories by quadratures.

Generalization of dissipated quantities in contact geometry.

Abstract

As a generalization and extension of our previous paper [Escobar-Ruiz and Azuaje, J. Phys. A: Math. Theor. 57, 105202 (2024)], in this work, the notions of particular integral and particular integrability in classical mechanics are extended to the formalisms of cosymplectic, contact and cocontact geometries. This represents a natural scheme to study nonintegrable time-dependent systems where only a part of the whole dynamics satisfies the conditions for integrability. Specifically, for Hamiltonian systems on cosymplectic, contact and cocontact manifolds, it is demonstrated that the existence of a particular integral allows us to f ind certain integral curves from a reduced, lower dimensional, set of Hamilton equations. In the case of particular integrability, these trajectories can be obtained by quadratures. Notably, for dissipative systems described by contact geometry, a particular integral can be viewed as a generalization of the important concept of dissipated quantity as well.