# [Physics] Type of stationary point in Hamilton’s principle

actionlagrangian-formalismvariational-principle

In this question it is discussed why by Hamilton's principle the action integral must be stationary. Most examples deal with the case that the action integral is minimal: this makes sense – we all follow the path with the least resistance.

However, a stationary point can be a maximal or minimal extremum or even a point of inflexion (rising or falling). Can anybody give examples from theory or practice where the action integral takes on a maximal extremum or a point of inflexion?

If you have in mind a mechanical system and if you are really referring to the Hamilton's principle (which tells how to find the actual trajectory the system follows between the fixed initial and final times at fixed initial conditions), then I doubt you can find such an example. The reason is that a mechanical system contains the kinetic term in its lagrangian, which is unbounded from above. That is, you can always construct a trajectory $\{x(t), {\dot x}(t)\}$ with as large action as you wish; so there is no maximum, not even a local one.