# Euler lagrangian equation associated with the variation of a given functional

• Raha
In summary, the Euler-Lagrange equation is a fundamental equation in the calculus of variations that is used in physics to find the extremum of a given functional. It is derived by setting the variation of the functional to be zero and solving for the unknown function. The difference between the Euler-Lagrange equation and the principle of least action is that the former is a necessary condition for an extremum while the latter states the path taken by a physical system is the one that minimizes the action. The Euler-Lagrange equation can be applied to any functional that depends on a single function, but for multiple functions a generalized form must be used. It is used in practical applications to derive equations of motion, solve optimization problems, and develop modern
Raha
Hi All,

is there anybody to give me some help on how I can calculate the Euler Lagrangian equation associated with variation of a given functional?
I am new with these concepts and have no clue about the procedure.

thanks a lot

## 1. What is the Euler-Lagrange equation and why is it important in physics?

The Euler-Lagrange equation is a fundamental equation in the calculus of variations that is used to find the extremum of a given functional. It is important in physics because it provides a mathematical framework for understanding the behavior of physical systems and is used to derive the equations of motion for a wide range of physical phenomena.

## 2. How is the Euler-Lagrange equation derived?

The Euler-Lagrange equation is derived by setting the variation of the functional to be zero and solving for the unknown function. This is done by applying the fundamental lemma of the calculus of variations and using the chain rule to simplify the resulting expression.

## 3. What is the difference between the Euler-Lagrange equation and the principle of least action?

The Euler-Lagrange equation and the principle of least action are closely related, but they are not the same. The Euler-Lagrange equation is a necessary condition for a function to be an extremum of a given functional, while the principle of least action states that the actual path taken by a physical system is the one that minimizes the action, which is the integral of the Lagrangian over time.

## 4. Can the Euler-Lagrange equation be applied to any functional?

Yes, the Euler-Lagrange equation can be applied to any functional that depends on a single function. However, for functionals that depend on multiple functions, such as in the case of field theories, a generalized form of the Euler-Lagrange equation known as the Euler-Lagrange equations of motion must be used.

## 5. How is the Euler-Lagrange equation used in practical applications?

The Euler-Lagrange equation is used in practical applications to derive the equations of motion for physical systems. It is also used in optimization problems to find the optimal solution for a given functional. Additionally, it is a key tool in the development of modern theories in physics, such as quantum mechanics and general relativity.

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