My question is not rigidly related to physics. The principle of least actions says that for any dynamical system there exists a function parameterized by $q$'s and $\dot{q}$'s such that the line integral of the function from state $A$ to state $B$ with respect to the parameter $s$ is stationary. Is there any rigorous mathematical proof of the principle?

Can I apply the principle of least action to study the evolution of a dynamical system in phase Space?