Variational Principles in Physics: Exploring Limits

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In summary, variational principles are a type of principle used in mechanics and optics. They are used to solve problems that cannot be solved with a simpler principle. There are other principles in physics that use this same type of approach, but they are not variational principles. Non-holonomic constraints are a type of problem that cannot be solved using a variational principle, and are instead solved with another principle. Finally, variational principles are not enough to describe mechanics - we also need the third law of motion.
  • #1
rushil
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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?
 
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  • #2
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
'non-holonomic'' ---- what's 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
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
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
The variational principle for entropy within the axiomatical formulation of equilibrium (classical/quantum) statistical physics is a famous example.

Daniel.
 
  • #7
Another example, from fluid mechanics, is Luke's variational principle for the irrotational flow&free surface problem
 
  • #8
rushil said:
'non-holonomic'' ---- what's 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!
http://en.wikipedia.org/wiki/Nonholonomic_system

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

https://www.amazon.com/gp/product/0486650677/?tag=pfamazon01-20
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
robphy said:
http://en.wikipedia.org/wiki/Nonholonomic_system
http://mitpress.mit.edu/SICM/
https://www.amazon.com/gp/product/0486650677/?tag=pfamazon01-20
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
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.
 

FAQ: Variational Principles in Physics: Exploring Limits

What are variational principles in physics?

Variational principles in physics are mathematical principles that describe the behavior of a physical system by minimizing or maximizing a certain quantity, such as energy or action. They provide a powerful tool for understanding the fundamental laws and equations governing physical systems.

How are variational principles used in physics?

Variational principles are used in physics to derive equations of motion, such as the famous Euler-Lagrange equations, which describe the dynamics of a system. They can also be used to find the equilibrium states of a system, or to determine the path that a system will follow between two points in space and time.

What are the advantages of using variational principles in physics?

One of the main advantages of using variational principles in physics is that they allow for a unified approach to understanding different physical phenomena. By using a single variational principle, one can derive equations that govern a wide range of systems, from classical mechanics to quantum mechanics.

Additionally, variational principles often lead to more elegant and concise equations and can provide deeper insights into the underlying physics of a system.

Are there any limitations to using variational principles in physics?

While variational principles are a powerful tool in physics, they do have some limitations. They may not always be applicable to systems with complex interactions or non-linear behavior. In some cases, alternative methods may be needed to accurately describe the behavior of a system.

Can variational principles be applied to other fields besides physics?

Yes, variational principles have been applied to various fields outside of physics, such as mathematics, economics, and engineering. They provide a general framework for optimization problems and can be used to find the most efficient or optimal solutions in these fields.

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