# Least Action Principle (Classical and Quantum Theory)

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"?

• Classical aspects of this question (v2) are discussed in this and this Phys.SE posts and links therein. Semiclassical aspects of this question (v2) are discussed in this and this Phys.SE posts and links therein. – Qmechanic Nov 27 '14 at 23:35