lagrangian formalism - More general invariance of the action functional

I will formulate my question in the classical case, where things are simplest.

Usually when one discusses a continuous symmetry of a theory, one means a one-parameter group of diffeomorphisms of the configuration space $M$ which fix the action functional $S:P\rightarrow \mathbb{R}$, where $P$ is the space of time evolutions, ie. differentiable paths in $M$. The idea is that, given some initial configuration $(x_0,v_0)\in TM$, there is a path in $P$ passing through $x_0$ with velocity $v_0$ and minimizing $S$ among all such paths. I will assume that this path is unique, which is almost always the case. Thus, if a diffeomorphism fixes $S$, it commutes with determining this path. One says that the physics is unchanged by taking the diffeomorphism.

Now here's the question: are there other diffeomorphisms which leave the physics unaltered? All one needs to do is ensure that the structure of the critical points of $S$ are unchanged by the diffeomorphism.

I'll be more particular. Write $P_{x_0,v_0}$ as the set of paths in $M$ passing through $x_0$ with velocity $v_0$. A diffeomorphism $\phi:M\rightarrow M$ is a symmetry of the theory $S:P\rightarrow \mathbb{R}$ iff for each $(x_0,v_0) \in TM$, $\gamma \in P_{x_0,v_0}$ is a critical point of $S|_{P_{x_0,v_0}}$ iff $\phi \circ \gamma$ is a critical point of $S|_{P_{\phi (x_0),\phi^* (v_0)}}$.

It is not obvious to me that this implies $S = S \circ \phi^{-1}$, where $\phi^{-1}$ is the induced map by postcomposition on $P$. If there are such symmetries, what can we say about Noether's theorem?

A perhaps analogous situation in the Hamiltonian formalism is in the correspondence between Hamiltonian flows and infinitesimal canonical transformations. Here, a vector field $X$ can be shown to be an infinitesimal canonical transformation iff its contraction with the Hamiltonian 2-form is closed. This contraction can be written as $df$ for some function $f$ (and hence $X$ as the Hamiltonian flow of $f$) in general iff $H^1(M)=0$. Is this analogous? What is the connection? It's been pointed out that this obstruction does not depend on the Hamiltonian, so is likely unrelated.

Thanks!

PS. If someone has more graffitichismo, tag away.

Answer

Let $\phi_s:P\rightarrow P$ be the induced diffeomorphism on the space of paths. You are assuming that the zero set $Zero(dS)$ coincides with the zero set $Zero(\phi_s^* dS)$. This does not even imply that $dS = \phi_s^* dS$, let alone $S = \phi_s^* S$.

An example would be a free particle on $\mathbf{R}$. Let $S=\int \dot{x}(t)^2dt$ and consider the scaling transformation $x\rightarrow\exp(s) x$. Then the critical points are straight lines $x(t)=x_0+v_0t$ and the transformation clearly preserves them. On the other hand, the action gets multiplied by $\exp(2s)$.

To understand the differences between $Zero(dS)=Zero(\phi_s^* dS)$ and $dS=\phi_s^* dS$, consider the graph of $dS$ in $T^*P$. The first condition only fixes the intersection points with the zero section, while the second condition fixes the graph itself. Clearly, in the $C^\infty$ world you can adjust the behavior of $dS$ away from the intersection points as much as you like. In the holomorphic world it would be enough to remember the Taylor expansions around the critical points.

Finally, $dS=\phi_s^*dS$ does not imply that $S=\phi_s^*S$: you only know that $S=\phi_s^*S + c(s)$, where $c$ is a locally-constant function on $P$ which vanishes at $s=0$.