Integral curves of derivations on locally semi-algebraic differential spaces
The reduction of symmetries of Hamiltonian systems in terms of differential spaces is studied. The definition of differential spaces is given and their properties are presented. Then, the closed differential subspaces of Euclidean and Hausdorff differential spaces that are locally diffeomorphic to closed subsets of Euclidean space are considered. The derivations on differential space structures are introduced and the associated integral curves are defined. A class of differential spaces which contains orbit spaces of proper actions of Lie groups on smooth manifolds is identified. The existence and uniqueness theorem for integral curves of derivations of smooth functions is proven for this class. A necessary and sufficient condition for a derivation to generate a local one-parameter group of local diffeomorphisms is also given. It is concluded that the results presented in this paper hold for all closed differential subspaces of Euclidean space and all differential spaces which are locally diffeomorphic to closed subsets of Euclidean space.
scientific article; zbMATH DE number 2046258