The Lagrangian method, also known as the calculus of variations, is a mathematical technique used to find the extrema (minimum or maximum) of a functional. The fundamental equation in the Lagrangian method is the Euler-Lagrange equation.

The Euler-Lagrange equation is derived from the principle of least action and is given by:

$$
\frac{\partial L}{\partial y} - \frac{d}{dx}\left(\frac{\partial L}{\partial y'}\right) = 0
$$

where L is the Lagrangian function, y is the dependent variable, y' represents the derivative of y with respect to the independent variable x, and ∂/∂y and ∂/∂y' denote partial derivatives with respect to y and y', respectively.

The Lagrangian function L typically depends on the variables y, y', and x, and it represents the integrand of the functional being optimized. The functional is typically an integral of the Lagrangian function over some interval.

The Euler-Lagrange equation is a necessary condition for a function y(x) to be an extremal (minimum or maximum) of the functional. Solving the Euler-Lagrange equation leads to the determination of the extremal function y(x) that satisfies the given boundary conditions.

In addition to the Euler-Lagrange equation, the Lagrangian method may involve the use of other mathematical techniques, such as integration by parts and applying constraints using Lagrange multipliers, depending on the specific problem being solved.
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