I am not sure if this is the right forum for this question, but I arrived at the question while studying the principle of stationary action so here it is: Consider the problem of finding the shortest path between two non-antipodal points on a sphere. Usually one solves this by using calculus of variations and finding trajectories where the distance integral is stationary. There are actually two solutions to this - the minor arc and the major arc of the great circle passing through the points with the minor clearly being the shorter one. Here is the thing that has been bothering me. I can see that the minor arc in addition to being stationary is the local minimum. However, I am having trouble trying to figure out if the major arc is a local maximum or local minimum or inflexion point. If I imagine adding a tiny bump to the path, I get a slightly longer path suggesting that the major arc must be a local minimum. However, if I imagine the path as a rubber band that I can continuous deform towards the minor arc, the length gets shorter suggesting a local maximum. Clearly I have a major gap in my understanding of stationary paths. Any idea about how I should think about this to resolve the confusion? Thanks P.S: Sorry if this has been asked before, but I searched through the site (and elsewhere) but could not find anything.