Least Action Principle (Classical and Quantum Theory)

Question

I) My first question would be "why should classical systems obey the principle of least action ?" When we find out the propagator in quantum physics, we find the amplitude to be equal to the sum over all possible paths, weighted by the phase $e^{iS/\hbar}$. Then one arrives at the conclusion that $\frac{\partial S[q]}{\partial q} $ must vanish when $\hbar \rightarrow 0$. So I'd presume the classical principle of least action is nothing but a special case of more generic quantum mechanical path-integral formulation. Is that fine?

II) Secondly, if we go through the derivation of how we arrive at the fact that the phase associated with a path is $e^{iS[q]/\hbar}$, starting from the first principle, we observe that the following assumptions play very crucial roles:

1. The momentum space is complete, i.e. $\sum |p\rangle \langle p| = 1 $.

2. The system is a Hamiltonian system,

   2a. The Hamiltonian is not exclusively time-dependent.

   2b. Kinetic energy operator is of the form $\hat{P}^2/2m$.

(Did I miss anything else?) Although many systems of interest easily follow these assumptions but I'm particularly curious about 2b. How would a system violating 2b behave under classical limit (can we even have such a system, first of all?), or what would happen to the least action principle.

So if (2) is all satisfied, which happens in most cases, then is it fine to say that "the principle of least action works because the phase space is a complete space"?

Answer 1

No, the principle of least action started well before quantum mechanics. It is a variational principle that, when applied to the action of a mechanical system, can be used to obtain the equations of motion for that system. The principle led to the development of the Lagrangian and Hamiltonian formulations of classical mechanics. this new formulation of mechanics in terms of hamiltonians led to a generalization, so every other branch of physics attempted to get the equations of motion from an adequate hamiltonian. In fact, most of modern physics reversed this and put the hamiltonian at the center of "existence", and the equations of motion as the consequences of H plus the least action principle. In the beginning was just a formalization trick that made the study of the dynamics of classical mechanical systems easier. See wikipedia for more details.