Orbits of complete families of vector fields on a subcartesian space are shown to be smooth manifolds. This allows a description of the structure of the reduced phase space of a Hamiltonian system in terms of the reduced Poisson algebra. Moreover, one can give a global description of smooth geometric structures on a family of manifolds, which form a singular foliation of a subcartesian space, in terms of objects defined on the corresponding family of vector fields. Stratified...

We report our recent results concerning integrability of Hamiltonian systems governed by Hamilton’s function of the form $H=1/2{\sum }_{i=1}^{n}p{²}_i+V\left(q\right)$, where the potential V is a finite sum of homogeneous components. In this paper we show how to find, in the differential Galois framework, computable necessary conditions for the integrability of such systems. Our main result concerns potentials of the form $V=V_k+V_K$, where $V_k$ and $V_K$ are homogeneous functions of integer degrees k and K > k, respectively. We present examples of integrable...