The action functionals in physics always seems weird to me. In this thread, I hope to formally justify them by arguing that for all kinds of dynamical systems, such magical functional exists.
Suppose $M$ is a smooth manifold, and $M \xrightarrow{f_t} M$ is a one-parameter family of diffeomorphism, with $f_0 = 1_M$. Denote $P(M)$ by the free path space of $M$.
Question 1. Does there exist a function $P(M) \xrightarrow{S} \mathbb{R}$ such that
- $S(\gamma_1 \ast \gamma_2) = S(\gamma_1) + S(\gamma_2)$, where $\ast$ denotes the concatenate operator and the $\gamma_i$'s are concatenatable.
- $\frac{\delta S}{\delta \gamma} |_{\gamma_0} = 0$ for all $\gamma_0$ traced out by $f_t$ near t=0 ?
If so, how large is the solution space for such $S$?
Question 2. Conversely, if no $f_t$ were given, but only the $S$ was given. For a point $x \in M$, does $S$ always induce a vector $v \in T_xM$ that dictates where $x$ is moving toward?