Small oscillations and the Heisenberg Lie algebra

Abstract

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

17 pages, it contains a theory about small oscillations in terms of the AKS scheme

References in corpus (2)

• p-Mechanics as a Physical Theory. An Introduction
• Relationships Between Quantum and Classical Mechanics using the Representation Theory of the Heisenberg Group

Cited by in corpus (3)

• The geodesic flow on nilmanifolds
• Invariant metrics and Hamiltonian Systems
• The Adler Kostant Symes scheme in physics