Euler - Lagrange Question

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.
  • #1


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Homework Statement

If L(y, y', x) = y[tex]^{2}[/tex] + y'[tex]^{2}[/tex] 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?
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  • #2
This is perfectly straightforward. Can you show us what you got and what the books answer is?
  • #3
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.

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