Euler Lagrange equation - weak solutions?

In summary, the conversation discusses the use of the Euler-Lagrange equation to find necessary conditions for a smooth function to be a minimizer. The question is raised if this approach can be extended to cover piecewise smooth solutions with a weak derivative. The suggestion is made to look for references on variational methods in formulating finite element approximations, although it is noted that there are other ways to create FE approximations.
  • #1
muzialis
166
1
Hello there,

I was wondering if anybody could indicate me a reference with regards to the following problem.

In general, the Euler - Lagrange equation can be used to find a necessary condition for a smooth function to be a minimizer.
Can the Euler - Lagrange approach be enriched to cover piecewise smooth solutions, with a weak derivative?

Any reference or hint would be so much appreciated.

Thanks

Muzialis
 
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  • #2
This is one way to create finite element approximations where the approximating function is piecewise smooth over the region (line, surface, or volume) covered by each element.

Look for something on variational methods of formulating finite elements.

Note, there are other ways to create FE approximations, which may appear to be mathematically "simpler", and avoid the difficuilty that for some applications of FE it is hard to find a variational form to minimise, but an "advanced" text on the math of FE methods should cover variational methods.

I learned this stuff a long time ago, so I can't give you a personal recommendation for a good modern textbook or website - sorry about that.
 

1. What is the Euler-Lagrange equation?

The Euler-Lagrange equation is a mathematical equation used in the field of calculus of variations. It is used to find the function that minimizes a certain functional, which is a function of a function.

2. What are weak solutions in the context of the Euler-Lagrange equation?

Weak solutions are solutions that satisfy the Euler-Lagrange equation in a generalized or weak sense. This means that the solution may not be differentiable at every point, but it still satisfies the equation in a weaker sense.

3. How is the Euler-Lagrange equation used in physics?

The Euler-Lagrange equation is commonly used in physics to find the equations of motion for a system. It is used to minimize the action, which is a functional that describes the motion of a system.

4. What is the significance of the Euler-Lagrange equation in optimization?

In optimization, the Euler-Lagrange equation is used to find the optimal solution for a given problem. It allows us to find the extrema of a functional, which can be applied to a wide range of optimization problems.

5. How is the Euler-Lagrange equation related to the calculus of variations?

The calculus of variations is a branch of mathematics that deals with finding functions that minimize or maximize certain functionals. The Euler-Lagrange equation is a fundamental tool in the calculus of variations and is used to find these minimizers or maximizers.

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