Necessary conditions for partial and super-integrability of Hamiltonian systems with homogeneous potentia

Abstract

We consider a natural Hamiltonian system of $n$ degrees of freedom with a homogeneous potential. Such system is called partially integrable if it admits $1<l<n$ independent and commuting first integrals, and it is called super-integrable if it admits $n+l$, $0<l<n$ independent first integrals such that $n$ of them commute. We formulate two theorems which give easily computable and effective necessary conditions for partial and super-integrability. These conditions are derived in the frame of the Morales-Ramis theory, i.e., from an analysis of the differential Galois group of variational equations along a particular solution of the system. To illustrate an application of the formulated theorems, we investigete three and four body problems on a line and the motion in a radial potential.