Limits of the variatonal principle

In summary, there are limits to the variational principle. Some theories can be reformulated as variational problems, but there may be some theories that cannot be.
  • #1
Umaxo
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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:)
 
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  • #2
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
 
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  • #3
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:)
 

1. What is the variational principle?

The variational principle is a mathematical principle that states that the path taken by a system between two points in time is the one that minimizes the action of the system. In other words, it is the path of least resistance or the path that requires the least amount of energy.

2. How is the variational principle used in science?

The variational principle is used in various fields of science, such as physics, chemistry, and engineering, to find the most efficient or optimal solutions to problems. It is also used to derive equations of motion and to understand the behavior of complex systems.

3. What are the limitations of the variational principle?

The variational principle only applies to systems that can be described by a Lagrangian function. It also assumes that the path of the system is continuous and differentiable, which may not always be the case in real-life systems. Additionally, it cannot be used for systems with constraints or singularities.

4. Can the variational principle be applied to quantum mechanics?

Yes, the variational principle can be applied to quantum mechanics through the use of the variational method. This method is used to approximate the energy of a quantum system by minimizing the expectation value of the energy with respect to a trial wave function.

5. How does the variational principle relate to the principle of least action?

The variational principle is closely related to the principle of least action, which states that the path taken by a system between two points is the one that minimizes the action of the system. The variational principle is a mathematical formulation of this principle and provides a more general framework for solving problems in science and engineering.

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