Principle of Least Action - not always valid?

[Master J]

It is stated in Landau & Lifgarbagez Mechanics that the Principle of Least Action is not always valid for the entire path of a system in phase space, but only for a sufficiently small segment of the path.

Can anyone expand on this?

How can we be sure that when we derive Lagrange's Equation that the principle is valid?

I guess the statement has slightly confused me - what are the consequences of this?

 

[atyy]

Basically, the equation gives a geodesic in some space. On a sphere, the great circles are geodesic. But between two points, you can go either direction on the geodesic, the short way or the long way. But between any two nearby points on the geodesic, it is always the short way. So some other quantity needs to be examined to figure out whether the extremal of the action we get is a minimum, saddle point, or maximum. The equation is almost always right, just the name "least action" isn't. Some specific examples are given by:
http://www.people.fas.harvard.edu/~djmorin/chap6.pdf
http://www.eftaylor.com/pub/Gray&TaylorAJP.pdf

The conditions under which differential equations can be derived from a variational principle are discussed by http://www.dic.univ.trieste.it/perspage/tonti/
http://arxiv.org/abs/1008.3177