Chris,
Thanks for the quotes.
Makes sense, since they are talking about coordinates and their rate of change. In that case, KE can certainly depend on ##{q}## and ##\dot{q}##. However, the way I see it, this only happens because the velocity itself depends on ##\dot{q}## and ##{q}##, either...
This is confusing. Are we saying that for the same velocity, just varying position can result in different KE?
I can see that KE can depend on position - but only because velocity depends on position - see AlephZero's example in post #5
I suppose it boils down to what we mean by velocity - if...
Hi AT,
Thanks for that explanation. The parameterization of the two curves using M1 and M2 really helps see the repetition along M1.
I think that resolves my question.
Once again thanks to Simon, AT and Orodruin. Now I can move on to page 8. of Landau Lifshitz Mechanics.
Thanks for the proof Orodruin!
After my response to AT, I went back and read some more about saddle points and it seems that by definition if you can find a smaller value and a larger value near a stationary point, it is a saddle point. So, by that definition it is a saddle point.
I was stuck...
Hi A.T.
Thanks. Initially, I did think that it might be a saddle point (and it may still be). However, when I try to visualize a surface with saddle point, I run into another problem
1. Either the surface has no global minimum - which not the case in our example.
OR
2. The surface has at...
Hi Simon,
Firstly, thanks for the response. Of course the greater arc is not a solution to the shortest path problem. However it is a solution to the stationary path problem.
I also understand that there is no global maxima - the big scribble you mention takes care of that.
My dilemma arises...
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...