Given a conservative mechanical system written in (for example) Newtonian form:
$m \ddot x= -\nabla U(x)$ (so the potential has only a "positional" dependence)
then we know that given the Lagrangian: $L(x,\dot x)=T-U \ $ then the admissible motions for the system are given by extremal points of the action of $L \ $.
Now, if we take the Hamiltonian of the Lagrangian (i.e. its Legendre transform) then we have the Hamilton's equations. I have three questions:
- Is it true that the admissible motions for the system are given by the solutions of Hamilton's equations?
- If $1$ is true, then is also true that every (Newtonian) conservative system is also volume preserving?
- This (for now only hypotetic) correspondence between Hamiltonian's solutions and admissible motions, is still valid if the potential is also velocity-dependent or time-dependent (i.e. $U=U(x,\dot x, t)$)?
P.s.I know that probably these are easy questions, but I have them because I've read Liouville preservation volume theorem and also I've read that the Hamiltonian satysfies the hypotesis of Liouville's theorem, and so I am quite impressed that every conservative mech. syst. is volume preserving.
Thank you in advance