\begin{definition}[Definition:Euler-Lagrange Equation]
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:
::$T X = \bigcup_{x \mathop \in X} \set x \times T_x X$
The ''Euler–Lagrange equation'', then, is given by:
:$\ds \map {L_x} {t, \map {\mathbf q} t, \map {\mathbf q'} t} - \dfrac \d {\d t} \map {L_v} {t, \map {\mathbf q} t, \map {\mathbf q'} t} = 0$
where:
:$L_x$ and $L_v$ denote the partial derivatives of $L$ with respect to the second and third arguments respectively.
\end{definition}