Euler - Lagrange Question

• WannabeNewton
In summary, the conversation is about finding the appropriate Euler Lagrange Equation for L(y, y', x) = y^{2} + y'^{2}. The person asking for help used the differential form but got a different answer from the one in the book. They eventually realized that they were treating the dynamical variables as regular variables, leading to mistakes.

Homework Statement

If L(y, y', x) = y$$^{2}$$ + y'$$^{2}$$ then find the appropriate Euler Lagrange Equation. I have absolutely no idea how to solve this. I used the differential form of the Euler Lagrange equations for a stationary action but the answer i got was nothing like the answer in the book so could anyone show me how to find the equation using the differential form?

This is perfectly straightforward. Can you show us what you got and what the books answer is?

Dick said:
This is perfectly straightforward. Can you show us what you got and what the books answer is?

Never mind I got it I keep forgetting the dynamical variables are actually functions and I keep treating them like variables in the equation. Stupid mistakes on my part. Sorry.

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1. What is the Euler-Lagrange equation?

The Euler-Lagrange equation is a mathematical equation used in the field of calculus of variations to find the extrema of a functional, which is a mathematical expression that takes in a function as its input and outputs a real number. It is named after mathematicians Leonhard Euler and Joseph-Louis Lagrange.

2. What is the significance of the Euler-Lagrange equation?

The Euler-Lagrange equation is significant because it provides a necessary condition for a function to be an extremum of a functional. This allows us to solve optimization problems in various fields such as physics, economics, and engineering.

3. How is the Euler-Lagrange equation derived?

The Euler-Lagrange equation is derived using the calculus of variations, which involves finding the stationary points of a functional. It is obtained by taking the functional's derivative with respect to the function and setting it equal to zero.

4. What is the relationship between the Euler-Lagrange equation and the principle of least action?

The Euler-Lagrange equation is closely related to the principle of least action, which states that a physical system follows the path of least action. This principle can be derived from the Euler-Lagrange equation by considering the functional to be the action of the system.

5. What are some practical applications of the Euler-Lagrange equation?

The Euler-Lagrange equation has many practical applications, including finding the optimal path for a spacecraft, minimizing energy consumption in electrical circuits, and optimizing the shape of a parachute for maximum air resistance. It is also used in the field of quantum mechanics to calculate the wave function of a system.