The algebraic entropy of classical mechanics

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 $\mathrm{1.82542377420108...}$, a fundamental constant associated to classical mechanics.

23 pages, 2 figures, submitted to J Math Phys