Abstract
•New integrable problems with additional weighted homogenous integral up to degree three in momenta have been introduced.•These problems are classified and physically interpreted by utilizing the Gaussian curvature.•We introduce a canonical transformation to convert our results to Fokker–Planck Hamiltonian related to stochastic model in the limit weak-noise.•We explain how we can use the concepts of mechanics in order to solve a Fokker–Planck equation.•A new integrable case for the swinging Atwood’s machine has been introduced and the additional integral has been announced.
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.