A Is Time Evolution in Phase-Space Equivalent to Canonical Transformations?

[Higgsono]

When we make a canonical transformation the volume in phase-space doesn't change. Likewise if we consider motion of a system in phase-space the volume won't change either according to Liouville's theorem.

Does that mean that every physical motion of the system in phase-space is equivalent to some parametrization of a set of canonical transformations?

 

[Twigg]

Yes and no.

No because the transformations change your coordinate system, not what physical state the system is in, whereas motion changes the state. Yes, because if you artificially kept the values of the coordinates constant, then yes you can come up with a series coordinate transforms which replicate the actual motion. It's the same mathematical machinery that let's you link together the rest frames of an accelerating observer via a one-parameter subgroup of Lorentz transformations and rotations.

 

[vanhees71]

Yes, for a Hamiltonian system time evolution is a canonical transformation ("flow") with the Hamiltonian as generator. For any phase-space function $f(q,p)$ you have
$$\dot{f}=\{f,H \},$$
because
$$\dot{f}=\dot{q}^j \partial_{q^j} f+\dot{p}_j \partial_{p_j} f=\partial_{p_j} H \partial_{q^j} f - \partial_{q_j} H \partial_{p_j} f=\{f,H\}.$$