Reduction of Hamiltonian systems with cyclic coordinates, and projectability in bundles

A Hamiltonian system on a smooth manifold which admits an algebra of integrals (noncommutative, in general) reduces, under some requirements, to a Hamiltonian system of less dimension. In this paper we show that this reduction can be described in terms of projecting in bundles. For simplicity we suppose that the algebra is commutative, therefore we deal with a Hamiltonian system with cyclic coordinates. We also consider the situation when the Hamiltonian is a quadratic form in the impulse coordinates, hence the Hamiltonian defines a metric on the configuration space, and, consequently, on the phase space.