In Landau-Lifshitz - Vol 1. Mechanics, right after the introduction of the principle of leas action, there is the following comment:

It should be mentioned that this formulation ($S = \int\limits_{t_1}^{t^2} L(q, \dot{q}, t) dt$) of the least action is not always valid for the entire path of the system, but only for any sufficiently short segment of the path. The integral for the entire path must have an extremum, but not necessarily a minimum. This fact, however, is of no importance as regards the derivation of the equations of motion, since only the extremum condition is used.

  1. Why is the least action principle not always valid for the entire path?
  2. Derivation of the equations of motion requires an extremum, not necessarily a minimum?
  3. How do we mathematically check if an extremum is in fact a minimum in this case?
  4. What are some physical examples that might give me further insight into this statement?