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.