Killing pairs and Newtonian integrals of motion
Killing pairs of symmetric tensor fields on a Riemannian manifold M describe integrals of the geodesics which are rational functions in the velocity coordinates. A well-known theorem of analytical mechanics states, that the orbits of a conservative Newtonian system, given by a Riemannian manifold (M,g) and a potential function V, which have energy E, can be parametrized such that they are normalized geodesics of the metric (E-V)g. The Killing pairs with respect to this metric therefore give integrals of the Newtonian motion, which are rational functions in the velocities. This allows a configuration space description of the integrals, closely parallel to the usual phase space formulation. The use of the configuration space language has attractive aspects in view of the intrinsic algebraic structure of the set of pairs of symmetric tensor fields.
rational functions in velocity coordinates
configuration space description of integrals
Killing pairs of symmetric tensor fields
scientific article; zbMATH DE number 3845106