#018: Express Liouville theorem for position and velocity
Category: Classical mechanics - Tags: Differential geometry, Riemannian geometry
Express the Liouville theorem of Hamiltonian mechanics in terms of position and velocity instead of position and momentum.
Mathematical problem. The Liouville theorem of Hamiltonian mechanics has its simplest expression in terms of conjugate quantities (i.e. position and momentum). In those cases where we can express momentum as a function of position and velocity (which corresponds to the existence of a Lagrangian), one should be able to express the same relationship in terms of position and velocity.
Given that \(p(x,v)\) will depend on the Hamiltonian, the expression may include a reference to the Hamiltonian (or the Lagrangian).
Physical significance. The Liouville theorem in physical terms means conservation of the number of states during time evolution. Position and conjugate momentum provide the best variables to keep that count, as the area using those units provides a direct count of states. However, conjugate momentum is not a directly measured quantity, as opposed to kinematic momentum.
Position and velocity (or position and kinematic momentum) provide a pair of variables that are measurable and therefore provide a better connection to the physics. However, they are not canonical and therefore the area they identify is not directly the count of states. It would be interesting to understand the expression of the theorem (and of the state count) in terms of these non-canonical but more physical variables.