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Subsection A.4.2 Calculus of Variations Formulation of the Euler-Lagrange Equation

Definition A.4.13 The Euler-Lagrange Equation.

For a functional $J[y] := \int_{x_1}^{x_2}{F\big(x,y(x),y_x(x)\big) \, dx}\text{,}$ the Euler-Lagrange Equation is

\begin{equation*} \frac{\partial F}{\partial x} - \frac{d}{dx} \big( \frac{\partial F}{\partial y_x} \big) = 0 \, . \end{equation*}

Remark A.4.14.

The Euler-Lagrange equation provides the necessary conditions for extremizing a path. However, it does not provide details on the nature of the extremum (e.g., if it is a minimum, maximum, or saddle point).

Remark A.4.15.

The stationary action paths derived from the Euler-Lagrange equation can alternatively be expressed in Hamiltonian phase space via Legendre transforms.

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