lagrangian formalism - Multiple classical paths from Hamilton's principle

Previous posts such as this ask about types of stationary point in Hamilton's Principle. There is, however, another aspect to discuss: the question as to whether the extremal path is unique.

One geometric way to envisage this is to assume that multiple paths are simultaneously extremal. I believe that this is an explanation for lenses, but I have not seen lenses explained as multiple classical solutions to Hamilton's Principle. (The multiple paths being the 360 degrees of rays between source and focus, etc also demonstrable through Fermat's principle.)

One can generalise lenses, but also consider a simpler case. Let the surface of a sphere be the action (phase space) surface which is minimized in classical paths. Thus (ignore antipodals here) between two points $A$ and $B$ the geodesic is the unique classical path. In quantum form the WKB approximation would no doubt have constructive maxima on this path.

However if the sphere has a disk (containing that geodesic) cut out, the shortest path now has exactly two choices: around one or the other rim from $A$ to $B$. Presumably WKB would maximize the quantum paths on these two (although I havent proved this). If so then classically we have a quantum-like phenomenon: a particle has a choice in going from $A$ to $B$. Experimentalists might see this and wonder whether the particle went from $A$ to $B$ via the LHS, the RHS or both....

Answer

[Another comment to answer transplant]

It seems like you're asking about a classical analog to the superselection sectors of quantum mechanics. One situation where this occurs in classical mechanics is when considering particle motion of a manifold with non-trivial topology - i.e. with holes and handles. In such cases there can be more than one extremal path from A to B. An example is an arcade pin-ball machine, if you're familiar with those.

Also, when you talk about optics it is important to keep in mind that there are two different regimes, those of wave optics and geometric optics. In the second case one has well-defined "trajectories" and you can find extremal trajectories. Not so in the first case.