Thanks for the anwser; I was trying to prove this for a covariant lagrangian; the only differencie I spot with your proof is that I have to write √(1-(u/c)^2)dτ instead of dt.julian said:Here I gave the proof for a simple example, should be easy for you to generalize it. Can give additional points if you like.
https://www.physicsforums.com/threads/what-is-a-lagrangian.823652/#post-5177331
The Euler-Lagrange equations are a set of differential equations that are used to find the extrema of a functional, which is a mathematical expression that takes in a function as its input and returns a number as its output. These equations were developed by mathematicians Leonhard Euler and Joseph-Louis Lagrange in the 18th century and are a fundamental tool in the field of calculus of variations.
The Euler-Lagrange equations are important because they provide a method for finding the optimal path or function that minimizes or maximizes a given functional. This is useful in various fields such as physics, economics, and engineering, where finding the most efficient or optimal solution is crucial.
To prove the Euler-Lagrange equations, you can follow the standard method of using the calculus of variations. This involves varying the function in the functional, then setting the variation to zero and solving for the function to find the critical points. The resulting equation will be the Euler-Lagrange equation for that particular functional.
The Euler-Lagrange equations have various applications in physics, such as in classical mechanics, quantum mechanics, and field theory. They are also used in economics to model optimization problems, in control theory to find optimal control strategies, and in image processing to find the best image restoration techniques.
Yes, there are some limitations to the Euler-Lagrange equations. They only apply to functions that have continuous first-order derivatives, and they do not always provide a unique solution. In some cases, there may be multiple solutions or no solutions at all. Additionally, the Euler-Lagrange equations cannot be used for non-smooth functions or functions with discontinuities.