# Classical trajectories that are not a minimum of the action [duplicate]

Are there physically realizable dynamical systems where the true trajectory is not a minumum action trajectory?

Formally, Lagrangian mechanics only requires that the trajectory be an extremum (or saddle point?), but all of the cases that I'm aware of, it is, in fact, a minimum. Are the other possibilities relevant for modelling any physical systems?

Not to mention, there are cases where the local extremum of the action isn't a physically realiziable path. Consider the plane with all the points satisfying $y > |x|$ removed. now, consider a start point of $(-2,1)$, and an end point of $(2,1)$ on this manifold, with the Lagrangian $\frac{1}{2}m\left({\dot x}^{2} + {\dot y}^{2}\right)$. Clearly, the minimum of the Lagrangian will take the particle on a path that leaves the manifold.