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Orbits of families of vector fields on subcartesian spaces

Jedrzej Śniatycki (2003)

Annales de l'Institut Fourier

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...

Oscillateur harmonique et opérateurs pseudodifférentiels

André Unterberger (1979)

Annales de l'institut Fourier

On donne des conditions larges sur un champ de normes symplectiques sur R 2 n pour que les opérateurs d’ordre zéro associés opèrent sur L 2 ( R n ) ; les éléments de cet espace se laissent alors écrire comme somme d’états propres, de niveau d’énergie bornée, de la famille d’oscillateurs harmoniques associée.

Overview of the differential Galois integrability conditions for non-homogeneous potentials

Andrzej J. Maciejewski, Maria Przybylska (2011)

Banach Center Publications

We report our recent results concerning integrability of Hamiltonian systems governed by Hamilton’s function of the form H = 1 / 2 i = 1 n p ² i + V ( q ) , 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...

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