The Lie algebra of classical mechanics

TL;DR

This paper studies the universal Lie algebra structure generated by kinetic and potential energies in classical mechanics, revealing its composition, dimensions, and a fundamental entropy constant, with implications for understanding mechanical systems.

Contribution

It characterizes the universal Lie algebra of classical mechanics, including its decomposition, dimension growth, and entropy, providing new insights into the algebraic structure of mechanical systems.

Findings

The Lie algebra decomposes into an abelian part and a freely generated algebra.

Dimensions of homogeneous subspaces grow exponentially with base approximately 1.8249.

Conjecture that systems with Euclidean kinetic energy are algebraically free.

Abstract

Classical mechanical systems are defined by their kinetic and potential energies. They generate a Lie algebra under the canonical Poisson bracket. This Lie algebra, which is usually infinite dimensional, is useful in analyzing the system, as well as in geometric numerical integration. But because the kinetic energy is quadratic in the momenta, the Lie algebra obeys identities beyond those implied by skew symmetry and the Jacobi identity. Some Poisson brackets, or combinations of brackets, are zero for all choices of kinetic and potential energy, regardless of the dimension of the system. Therefore, we study the universal object in this setting, the `Lie algebra of classical mechanics' modelled on the Lie algebra generated by kinetic and potential energy of a simple mechanical system with respect to the canonical Poisson bracket. We show that it is the direct sum of an abelian algebra $\mathcal X$, spanned by `modified' potential energies isomorphic to the free commutative nonassociative algebra with one generator, and an algebra freely generated by the kinetic energy and its Poisson bracket with $\mathcal X$. We calculate the dimensions $c_n$ of its homogeneous subspaces and determine the value of its entropy $\lim_{n\to\infty} c_n^{1/n}$. It is $1.8249\dots$, a fundamental constant associated to classical mechanics. We conjecture that the class of systems with Euclidean kinetic energy metrics is already free, i.e., the only linear identities satisfied by the Lie brackets of all such systems are those satisfied by the Lie algebra of classical mechanics.