There is 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 systèmes dynamiques, Dunod, Paris, 1969.

J.-M. Souriau, La structure symplectique de la mécanique décrite par Lagrange en 1811, Mathématiques et sciences humaines, tome 94 (1986), pages 45–54. Numérisé par Numdam, http://www.numdam.org.

https://www.google.com/url?sa=t&source=web&rct=j&url=http://marle.perso.math.cnrs.fr/diaporamas/ManifoldMotionsMass.pdf&ved=2ahUKEwjjt5SWz8LsAhXKMewKHfg-D5wQFjAGegQIBxAB&usg=AOvVaw00m-2HERW5vWi10i1m3_hd

Foundations of Mechanicsby Abraham/Marsden? This book seemed to be fairly well known when I was in college, especially when the greatly expended 1980 2nd edition was published. $\endgroup$ – Dave L Renfro Oct 19 at 19:57