The algebraic entropy of classical mechanics

TL;DR

This paper introduces a polynomially graded Lie algebra model for classical mechanics, calculates the dimensions of its homogeneous subspaces, and determines its algebraic entropy as a fundamental constant.

Contribution

It defines a new class of Lie algebras for classical mechanics, provides an algorithm for dimension calculation, and computes the algebra's entropy value.

Findings

The Lie algebra of classical mechanics is polynomially graded.

An explicit algorithm for calculating subspace dimensions is provided.

The algebraic entropy is approximately 1.8254, a fundamental constant.

Abstract

We describe 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. It is a polynomially graded Lie algebra, a class we introduce. We describe these Lie algebras, give an algorithm to calculate the dimensions $c_n$ of the homogeneous subspaces of the Lie algebra of classical mechanics, and determine the value of its entropy $\lim_{n\to\infty} c_n^{1/n}$. It is $1.82542377420108...$, a fundamental constant associated to classical mechanics.