I have seen the principle of least action also being described as stationary action. I can see that the calculus is searching for a stationary point which could be a minimum, maximum, or saddle. However, given any two fixed points there are always infinite paths between them; hence no definable maximum path length. Although there will always be a minimum path. Is that correct? Why stationary action if the maximum is indeterminable? Also, I have seen the least action path solved in one dimension but what happens with more degrees of freedom? I could imagine that multiple paths would yield the same minimum value?