Dynamical algebras in classical mechanics (Q1309015)

The idea of dynamical algebras (or groups) in quantum theory which are not symmetry algebras, is that the physical quantities determining the quantum numbers of the system form an algebra or an enveloping algebra, the Hamiltonian being a function of these operators. More generally a covariant wave equation involving this algebra of operators is written down. Hence the Hilbert space of states can be constructed from the irreducible representations of this algebra. Thus spectrum and transitions can be calculated without an explicit internal dynamics. Usually this algebra is universal, different systems belonging to different realizations or representations of this algebra on more and more complicated spaces. This idea is here formulated for classical mechanics. A collection of observables are assumed to form a Lie algebra under Poisson brackets. If \(H\) is a function of these observables the dynamics simplifies. The coadjoint orbits, which are symplectic manifolds, correspond to all possible values of these observables. On each orbit there is a reduced \(H\) which determines the time evolution of all observables.