Understanding the Principle of Least Action in Multidimensional Systems

[Nick.]

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?

 

[anorlunda]

It is not entirely clear what you are asking. But a stationary point is by definition at either a minimum or a maximum. Since there is no maximum in your case, it must bea minimum. Does that address your question?

 

[Lisa_304]

hi nick,, I agree with anorlunda you should be more specific next time. I try to understand your questions and I hope to answer to you and solve some of your questions.

So you have the action which is S=∫ L dt ,where L is the lagrangian.

The Lagragian may be L=1/2 m r˙^2 - mgr , where r˙ is a the magnitude of velocity.
r could be r˙=x˙^2+y˙^2+z˙^2
r˙=r˙^2+(r*θ˙)^2

Don't worry about three or two dimensional problems, you simply find your position vector in a random position and then you find the derivative which is the velocity and then you find the magnitude of velocity.Furthermore about minimum and maximum...you always ask this action integral to be minimum, because you always want your action to be done using the least energy. That's how nature works, when nature wants to do something chooses the path that needs the lowest energy to accomplish the action. Don't forget that maths always describe nature.If I understood well you also ask what happens to the paths..if you have a problem with more than one dimension..Don't analyse it too much especially if you learn these stuff in a physics course..What I consider as important is that when you set dS=0 and doing some math (that you could find them in almost every classical mechanics book) and you end up with euler's langrange's equations that give you the proper equation of motion.
From all these virtual paths E-L equations choose for you the right path. That's all in my opinion. Now if you are a mathematician then okay maybe you should go deeper than that.

I hope to help you my friend. :)