I Limits of the variatonal principle

[Umaxo]

Hi,

I got curious on the limits of variatonal principle. As far as i know all of the theories can be reformulated as problem of finding extremum of some action. Not only that, but it seems to be most convenient method for looking for new theories in a lot of cases.

So my question is, what are the limits of it? Can any concievable theory be reformulated into variatonal problem with scalar lagrangian? Or the set of such theories is limited, and we are just lucky that so far nature seems to like this principle? Is there some good (mathematical) textbook dealing with this limits of variatonal principle?

Thanks:)

 

[jedishrfu]

While I can't comment on your question directly, I do know it's often true that we have a variety of means to compute answers in Physics. The approach has been to pick the most straightforward way until you run into a wall of difficulty and then you look for a more high-powered way to solve it. By straight forward I mean how we were first taught to solve it.

In Susskind's book on the principle of least action he derives it using a variation method and then shows that its equivalent to and nothing more than F=ma in disguise.

From that we see that we can solve many problems using the Newtonian force model but when we hit a wall of difficulty we can use the more high-powered but elegant and mysterious Lagrangian model.

Here's some lecture notes on it from Prof DANIEL D. BAUMANN:

http://www.damtp.cam.ac.uk/user/db275/LeastAction.pdfmore interesting topics from Prof Baumann here:

http://www.damtp.cam.ac.uk/user/db275/concepts.pdf

 

[Umaxo]

Thanks for the links.

I recently grabbed a book on mechanics from Goldstein, Poole and Safko as suplement for Landau-Lifshitz and it seems that indeed there are even mechanical systems that cannot be modeled using variational principle. Goldstein starts with Newtons force law which he wants to use for describing systems with constrains. And from what i gathered, some constraints indeed cannot be modeled like this.

Perhaps after going through the whole material and digesting it i get my answer:) If not, i will be back:)