From group-theory perspective, what is Action? I'm trying to self-study group theory in relation to physics, so I apologize in advance if I'm missing something obvious here or if the question is not clear.

We know there is a symmetry group, which can be either Galilean group or Poincare group, acting on physical space: a set of transformations that change $(t,x,y,z) \to (t',x',y',z')$ in a certain way.
However, what we need study is how they act on the space of States, and States (in classical mechanics) include momenta. The relation between, say, $p^x$ and $(t,x,y,z)$ comes from the Action $A(t,x,y,z)$:
$$
  p^x = -\partial A(t,x,y,z)/\partial x \tag{1}
$$
where $A(t,x,y,z)$ is the Action.
So there's this additional thing called Action which comes into play. However, and it seems to me to be the key point, the Action cannot be chosen arbitrarily: it has to be constructed in such a way that a transformation from our group turns a solution of E-L equations into a solution of E-L equations with the same Action.
If we represent $A$ in the functional form:
$$
A[x(t), y(t), z(t)] = \int L(t,x(t),y(t),z(t),\dot{x}(t), \dot{y}(t), \dot{z}(t)) dt\tag{2}
$$
this condition implies that there exists a function $F(t,x,y,z)$, s.t.
$$
L(t,x(t),y(t),z(t),\dot{x}(t), \dot{y}(t), \dot{z}(t)) dt - L(t', x'(t'),y'(t'),z'(t'),\dot{x}'(t'), \dot{y}'(t'), \dot{z}'(t')) dt' = dF\tag{3}
$$
where primed values are related to unprimed values by our transformation.
One can check that given a particular group, it is possible to recover standard forms of galilean or relativistic actions (up to constants). For example, assuming homogeneity and isotropy of space and time, we can assume that
$$
L(t,\vec{x}, \vec{v}) = \mathscr{L}(\vec{v}^2)\tag{4}
$$
and then taking an infinitesimal transformation from Poincare group:
$$
t' = t - (\vec{u},\vec{x})/c^2\tag{5}
$$
$$
\vec{x}' = \vec{x} - \vec{u}t\tag{6}
$$
we conclude that this condition can only be satisfied if
$$
\frac{\mathscr{L}}{c^2} + 2\left(1-\frac{\vec{v}^2}{c^2}\right)\mathscr{L}' = \textrm{const}\tag{7}
$$
Solving it, we get
$$
\mathscr{L} = \textrm{const}\times c^2 + \textrm{const}_2\times\sqrt{1-\frac{\vec{v}^2}{c^2}}\tag{8}
$$
To recover the standard form just set $\textrm{const} = 0$, $\textrm{const}_2 = -mc^2$.

So, we have this full-differential condition which is some sort of condition on $L$ and hence on $A$, arising from the group we study.
My question is: from the group-theory perspective, what is this condition? What can we learn about $L$ (or $A$) given just the group, without explicitly solving the equation above?
 A: *

*First of all, now that OP mentions group theory, it should probably be stressed that group action in mathematics has nothing to do with action in physics.


*In physics, there are various types of actions, cf. e.g. my related Phys.SE answers here & here. E.g. OP's eq. (2) is an action functional, while the action in eq. (1) is not.


*The condition (3) is known as a quasi-symmetry.


*Concerning a Poincare invariant action principle (8) for the massive point particle, see e.g. this, this & this related Phys.SE posts.


*Concerning classification of symmetries for an action, see e.g. this &  this related Phys.SE posts.
A: In the following answer I will discuss three instances of the action concept. There are recurring patterns, and I expect that those recurring patterns will give you clues.
Three categories of cases where an 'action concept' is used are:
-Optics
-Statics
-Dynamics
In geometrical optics the quantity that is used as the 'action' for that context is the duration of the path that the light takes (Fermat stationary time).
The path that the light takes is recovered by taking the derivative of the action. The time itself is not a factor in the following sense: what counts is the derivative of the time. That is: there is only one way to recover the path that the light takes: by taking the derivative of the action.
Taking the derivative is necessary, and sufficient.
We have the following well known case: a reflecting rim in the shape of an ellipse, with a light source at one focus of that ellipse. As we know: with that setup all the paths of from one focus to the other have the same duration.
We can designate the case of ellipse-shaped-rim-with-light-source-at-one-focus as a critical case. The critical case points out a demarcation between classes of cases: cases below and above the critical case.
Below:
The simplest case is reflection against a flat mirror: for the actual path the derivative of the time is zero, and it's a minimum.
Above:
Set the reflecting surface up to be more curved than in the ellipse-shaped case. Interestingly, in that case the point of zero derivative of the time corresponds to a maximum.

This is a recurrent pattern, and I believe it is key to the action concept.


In statics: take the case of the catenary problem. You set up an expression for the total potential energy. In the category of Statics the total potential energy is the corresponding action. To solve for the shape of the catenary we impose the condition that the derivative of the potential energy with respect to the height coordinate is zero.
Mathematically: the condition that the derivative of the potential energy must be zero allows two solutions. One with the center of mass below the anchor points (hanging chain) and one with the the center of mass above the anchor points (catenary arch.)


Hamilton's stationary action has the same pattern as in the case of Fermat's stationary time: there is a critical case, pointing out demarcation between classes of cases.
For Hamilton's stationary action the critical case is harmonic oscillation, evaluated for half a period of the oscillation (or an integer multiple of that). As we know, with Hooke's law the potential increases quadratic with displacement.
In that specific case, the critical case: all amplitudes of the oscillation have the same action. With harmonic oscillation the potential energy and kinetic energy are both quadratic, and they are equal in magnitude, so for Hamilton's action the two drop away against each other for every amplitude of the oscillation.
This leaves the trajectory underdetermined. In order to narrow down to a single trajectory some initial condition has to be supplied.
In the case of Hooke's law the potential increases quadratic with the displacement.
Below that critical case are the classes of cases where the potential increases slower than that. Then the true trajectory corresponds to a minimum of Hamilton's action.
Above that critical case are the classes of cases where the function for the potential is of higher order than the function for the kinetic energy. For instance, when the potential increases in proportion to the cube of the displacement the true trajectory corresponds to a maximum of Hamilton's action. (To be clear: this is not an expectation; I verified that.)


The recurring pattern:
-For each context there is a bespoke action.
-To recover the required information you have to take the derivative of the action, looking for the point where that derivative is zero.
-Whether the point where the derivative is zero is a minimum or a maximum is immaterial.
(I believe the interpretation of the above properties is fairly straightforward. However, I prefer to give people opportunity to draw conclusions on their own. So I won't proceed to that interpretation here.)
So:
For expressing the action concept in terms of group theory you will have to accomodate the above properties.



I rather glossed over Fermat stationary time. Let me expand on that a little here.
As we know, reflection and refraction are about angles. So: how does taking the derivative of the Fermat time relate to angles? In the case of reflection of light the velocity is always the same. That means the length of the Fermat path and the Fermat time are in direct proportion to each other. Taking the derivative of the time is also the derivative of the length of the Fermat path. The derivative is a ratio of two numbers, and an angle is a ratio of two numbers: sine/cosine. That is how the reflection angle is recovered.
Refraction adds the dimension that the speed of light is a function of the optical density of the medium. In the end it works out the same: the derivative of the Fermat time recovers Snell's law because taking the derivative recovers the angles.
