Dynamical algebras in classical mechanics

For a spectrum-generating algebra of classical observables, it is proven that the phase space dynamics simplifies to a Hamiltonian system on submanifolds of the algebra's dual. These submanifolds are coadjoint orbits if the algebra arises from a symplectic group action. If the Hamiltonian splits into the sum of a function of the algebra generators plus a commuting part, then the dynamics transfers to the dual space and an explicit formula is given for the flow vector field on the coadjoint orbits. A unique feature of the presentation is that all constructions are at the Lie algebra level.