Small oscillations on and Lie theory

making use of lie theory we propose a model for the simple harmonic oscillator and for the linear inverse pendulum of . in both cases the phase space are orbits of the coadjoint representation of the heisenberg lie group. these orbits and the heisenberg lie algebra are included in a solvable lie algebra admitting an ad-invariant metric. the corresponding quadratic form induces the hamiltonian and the associated hamiltonian system is a lax equation.