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

As a generalization and extension of our previous paper (Escobar-Ruiz and Azuaje 2024 J. Phys. A: Math. Theor. 57 105202), 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 framework for studying dissipative systems, enabling a reduction of the equations of motion and, in certain cases, allowing explicit solutions to be found within a subset of the overall dynamics where integrability conditions are met. Specifically, for Hamiltonian systems on cosymplectic, contact and cocontact manifolds, it is demonstrated that the existence of a particular integral allows us to find certain integral curves from a reduced, lower dimensional, set of Hamilton's 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. [ABSTRACT FROM AUTHOR]