Small oscillations and the Heisenberg Lie algebra

TL;DR

This paper uses the Adler-Kostant-Symes scheme to analyze small oscillations in systems modeled by the Heisenberg Lie algebra, revealing conditions for integrability and explicit solutions for certain Hamiltonian systems.

Contribution

It connects the integrability of quadratic Hamiltonian systems on coadjoint orbits to derivations of the Heisenberg Lie algebra, providing a Lie algebraic framework for analyzing oscillatory systems.

Findings

Complete integrability linked to abelian subalgebras of derivations in the Heisenberg algebra.

Explicit solutions for harmonic oscillator systems near equilibrium.

Poisson commutativity related to derivation commutativity in the 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.