# Variational Principles

1. Dec 13, 2005

### rushil

By variational principles I mean e.g. the Principle of Least action in Mechanics and Fermat's principle in Optics.
Which are the other such principles in Physics, Can our entire knowledge of physics be described in terms of just the variational principles???

2. Dec 13, 2005

### Physics Monkey

In general, no. For example, in classical mechanics, there are many problems that involve frictional forces or non-holonomic constraints which cannot generally be handled with a variational principle.

3. Dec 13, 2005

### rushil

'non-holonomic'' ---- whats that!!??!?!?!?!?

Can you please mention other variational principles in Physics!! Also, maybe you are talking about an observation that is not ideal -- of course I am talking about ideal cases!! Friction is not exactly an ideal case of Newton's laws!!

4. Dec 13, 2005

### Physics Monkey

I can't find a good website on non-holonomic constraints at the moment, but basically anytime the constraint can expressed as a relation amongst the coordinates, it is holonomic. Everything else is non-holonomic, for example, the rolling constraint is an example of non-holonomic constraint because it relates velocities rather than coordinates.

Also, you did ask if our "entire" knowledge of physics can be stated in terms of variation principles. This isn't possible as I indicated. The famous infinities of quantum field theory are another example of a situation where additional physical insight beyond the physics contained in the action is required.

5. Dec 13, 2005

### rushil

Can yo please mention a site which mentions variational principles known till date!!

Also, I want to tell you that my question is inspired by a chapter on a similar topic in the Feynman Lectures... he showed that Newton's Second Law is equivalent to a variational princeiple in Energy - The Lagrange Principle --- but there again the variational principle does not imply in any way the Third law - the third law is an independent system! So the Largrange Principle is itself not sufficent to describe Mechanics - we need the Third law as well!! Am I right????

6. Dec 14, 2005

### dextercioby

The variational principle for entropy within the axiomatical formulation of equilibrium (classical/quantum) statistical physics is a famous example.

Daniel.

7. Dec 14, 2005

### arildno

Another example, from fluid mechanics, is Luke's variational principle for the irrotational flow&free surface problem

8. Dec 14, 2005

### robphy

http://en.wikipedia.org/wiki/Nonholonomic_system

http://mitpress.mit.edu/SICM/

https://www.amazon.com/gp/product/0486650677/
The Variational Principles of Mechanics (Dover Books on Physics and Chemistry) - by Cornelius Lanczos
If I recall, this discusses a bunch of minimization principles associated with Maupertuis, Hertz, Fermat, etc...

Here's something in Fluid Mechanics
http://www-users.york.ac.uk/~ki502/review6.pdf

9. Dec 14, 2005

### arildno

As a pendant to the V.I Arnold reference in fluid mechanics, I'd like to mention Chandrasekhar's monograph on, among other issues, hydrodynamic stability, where he advocates the use of a variational approach. I think it was written a few years prior to Arnold's work, but I haven't studied this in any detail.