In classical mechanics, the least action principle states that the real path linnking $(x_1, t_1)$ and $(x_2, t_2)$ is an extremal of the action functional.
The question is, how many such solutions are there? For a free particle, apparently there is only one. For a harmonic oscillator, it seems that there is also only one solution.
But how about a general system? Could anyone give an example with two or even more solutions?
A simple system with infinitely many solutions: a mass orbiting around a fixed gravity potential well. Starting at a certain radius, the mass can orbit the gravitational center with a circle along any direction. They will all reach the opposite end at the same time. Symmetry is probably a general way to have multiple such solutions with specific end points.
A non-symmetry example: a ball bouncing against a wall (a very steep potential wall, without dissipation of course). Given two points in space and time, the ball can go straight to the end with slow speed, or bounce on the potential wall and then go the end point with fast speed. Both trajectories can reach the same end point (same space and time). Symmetry seems to play no role here, as far as I can tell.
In general, The number of solutions depends on specific situations. For instance, two space-like events in special relativity has no solution, as far as I know. Well, no "physical" solution I guess.
From my understand, principle of least action is just a tool to get the equation of motion in classical physics. One doesn't really need to think about varying the path as a minimization problem in most situations. The exceptions might be path integral in quantum mechanics or optics in changing median.
There is no good answer to this question. Depending on the particulars of the problem you are trying to solve, there may be many extremal solutions that maximize or minimize the action. The key is that we typically look for solutions that are reasonably close to either some starting configuration we know to be physically realistic, or else solutions that satisfy some other metrics of “physical” solutions. In reality, there may not be any obvious way to know you have rigorously sampled a phase space to the point that you have taken every relevant path. One relevant point is that some paths may be candidates in an idealized scenario, but incur, for instance, more resistance in situations where non ideal behavior is incorporated (friction, etc.). This is actually quite akin to a problem in computational chemistry where it is never possible to 100% assure that you have found the true minimum energy structure of a molecule. You can’t assure that you have sampled the degrees of freedom sufficiently to do any more than say you seem to have found a reasonable candidate minimum energy structure.
