Definition:Euler-Lagrange Equation

Definition
The Euler–Lagrange equation is an equation satisfied by a function $\mathbf q$ of a real argument $t$, which is a stationary point of the functional:


 * $\ds \map S {\mathbf q} = \int_a^b \map L {t, \map {\mathbf q} t, \map {\mathbf q'} t} \rd t$

where:
 * $\mathbf q$ is the function to be found:
 * $\mathbf q: \closedint a b \subset \R \to X : t \mapsto x = \map {\mathbf q} t$

such that:
 * $\mathbf q$ is differentiable
 * $\map {\mathbf q} a = \mathbf x_a$
 * $\map {\mathbf q} b = \mathbf x_b$
 * $\mathbf q'$ is the derivative of $\mathbf q$:
 * $\mathbf q': \closedint a b \to T_{\map {\mathbf q} t} X: t \mapsto v = \map {\mathbf q'} t$


 * $T_{\map {\mathbf q} t} X$ denotes the tangent space to $X$ at the point $\map {\mathbf q} t$


 * $L$ is a real-valued function with continuous first partial derivatives:
 * $L: \closedint a b \times T X \to \R: \tuple {t, x, v} \mapsto \map L {t, x, v}$

where:
 * $T X$ is the tangent bundle of $X$ defined by:
 * $\ds T X = \bigcup_{x \mathop \in X} \set x \times T_x X$

The Euler–Lagrange equation, then, is given by:
 * $\map {L_x} {t, \map {\mathbf q} t, \map {\mathbf q'} t} - \dfrac \d {\d t} \map {L_v} {t, \map {\mathbf q} t, \map {\mathbf q'} t} = 0$

where:
 * $L_x$ and $L_v$ denote the partial derivatives of $L$ with respect to the second and third arguments respectively.

Also known as
The Euler-Lagrange equation is often taken in the plural: Euler-Lagrange equations, as it actually defines a system of differential equations.

They are sometimes referred to as Lagrange's equations of motion.