There is a a "symplectic structure" on the set of body motions.
During the years 1960–1970, Jean-Marie Souriau, proved that under very general assumptions, the set of all possible solutions of a classical mechanical system, involving material points interacting by very general forces, has a smooth manifold structure (not always Hausdorff) and is endowed with a natural symplectic form. He called it the manifold of motions of the mechanic
J.-M. Souriau, Structure des syst`emessystèmes dynamiques, Dunod, Paris, 1969.
J.-M. Souriau, La structure symplectique de la m´ecanique d´ecritemécanique décrite par Lagrange en 1811, Math´ematiquesMathématiques et sciences humaines, tome 94 (1986), pages 45–54. Num´eris´eNumérisé par Numdam, http://www.numdam.org.