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TimeRip496
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I am trying to derive it but I am stuck at the boundary condition. What is this boundary comdition thing such that the value must be zero?
Yes.robphy said:Can you elaborate?
Is this related to the variation being zero at the endpoints?
TimeRip496 said:Yes.
The Euler Lagrange equation is a mathematical expression used in the field of calculus of variations to find the extremum of a functional. It is used to find the function that minimizes or maximizes the functional, representing a physical quantity, over a specified interval.
The Euler Lagrange equation is essential in many areas of physics, engineering, and mathematics. It is used to find the motion of a system in classical mechanics, the path of a light ray in optics, and the shape of a soap film in surface tension, among others. It also provides a general framework for solving optimization problems in various fields.
The boundary conditions in the Euler Lagrange equation specify the values of the function at the endpoints of the interval. These conditions are necessary for finding a unique solution to the equation and determining the extremum of the functional.
The Euler Lagrange equation is derived by setting the derivative of the functional with respect to the function equal to zero and then applying the fundamental theorem of calculus. This results in a differential equation that can be solved to find the function that minimizes or maximizes the functional.
Yes, the Euler Lagrange equation can be extended to higher dimensions, known as the Euler-Lagrange PDE. It is used to find the extremum of a functional in multiple variables and is commonly used in fields such as fluid dynamics, electromagnetism, and quantum mechanics.