First integrals of motion for two dimensional weight-homogeneous Hamiltonian systems in curved spaces

Abstract

In our work (Nonlinear Dyn. 93: 933–943, 2018), we introduced the necessary conditions for the integrability for certain type of Hamiltonian systems by investigating the properties of the differential Galois group of the normal variational equations along a certain particular solution. The current work complements this study by presenting the sufficient conditions of the integrability by constructing the weighted homogeneous first integral of motion that is independent on the Hamiltonian. The obtained integrable problems have been classified by utilizing the Gauss curvature. Some of them describe physically the motion in the Euclidean plane, on a standard sphere, on pseudo-sphere, and on an unspecified surface having variable Gaussian curvature. We present certain canonical transformation converting the obtained new integrable systems to a Fokker–Planck Hamiltonian associated to a Fokker–Planck equation that corresponds to a certain limit weak-noise stochastic model. Moreover, we elucidate how we can use the mechanical concepts to solve a Fokker–Planck equation. The existence of extra parameters in the obtained new integrable systems enables us to identify and generalize some of the previous results for certain values of those parameters. Consequently, we introduce a new integrable case for a swinging Atwood’s machine on a zero-level of the energy.