Small oscillations and the Heisenberg Lie algebra

The Adler - Kostant - Symes scheme is used to describe mechanical systems for quadratic Hamiltonians of R-2n on coadjoint orbits of the Heisenberg Lie group. The coadjoint orbits are realized in a solvable Lie algebra g that admits an ad-invariant metric. Its quadratic induces the Hamiltonian on the orbits, whose Hamiltonian system is equivalent to that on R-2n. This system is a Lax pair equation whose solution can be computed with the help of the adjoint representation. For a certain class of functions, the Poisson commutativity on the coadjoint orbits in g is related to the commutativity of a family of derivations of the (2n+1)-dimensional Heisenberg Lie algebra h(n). Therefore the complete integrability is related to the existence of an n-dimensional abelian subalgebra of certain derivations in h(n). For instance, the motion of n-uncoupled harmonic oscillators near an equilibrium position can be described with this setting.