  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.
