Note on integrability of certain homogeneous Hamiltonian systems

Abstract

In this paper we investigate a class of natural Hamiltonian systems with two degrees of freedom. The kinetic energy depends on coordinates but the system is homogeneous. Thanks to this property it admits, in a general case, a particular solution. Using this solution we derive necessary conditions for the integrability of such systems investigating differential Galois group of variational equations.

Transformation of the Stäckel matrices preserving superintegrability

“…In similar manner we can get infinite families of superintegrable systems associated with superintegrable systems with masses depending on coordinates from [11,35,40]…”

Superintegrable systems with a position dependent mass: Kepler-related and oscillator-related systems

“…Second, quadratic superintegrability is a property very related with Hamilton-Jacobi (H-J) multiple separability (Schrödinger separability in the quantum case) and this property is also true for systems with a position dependent mass. This question (H-J separability approach to systems with a pdm) was studied in [9] (in this case the pdm depends on a parameter κ) and more recently in [10] (in this last case the pdm Hamiltonians studied were also related with those recently obtained through a differential Galois group analysis in [1]).…”

“…It is known that the Liouville formalism characterize the Hamiltonians that are integrable but it does not provide a method for obtaining the constants of motion; therefore it has been necessary to carry out several different methods for searching integrals of motion (Noether symmetries, Hidden symmetries, Lax pairs formalism, bi-Hamiltonian structures, etc). In a recent paper Szuminski et al studied [1] families of Hamiltonians of the form H nk = 1 2 r n−k p 2 r + p 2 φ r 2 + r n U (φ) , (n y k are integers) and then, making use of some previous results of Morales-Ruiz and Ramis related with the differential Galois group of variational equations [2,3,4], they derive necessary conditions for the integrability of such systems. Then using some rather involved mathematics (related with the hypergeometric differential equation) they arrive to a certain number of Hamiltonians and prove that four of them, given by…”

A note on some superintegrable Hamiltonian systems

“…A superintegrable system may occur in some other context and not be written in any recognisable form, so requires some judicious change of coordinates. In [7] the authors considered Hamiltonian functions of the form…”