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:


 * $\displaystyle S \left({\mathbf q}\right) = \int_a^b L \left({t, \mathbf q \left({t}\right), \mathbf q' \left({t}\right)}\right) \, \mathrm d t$

where:
 * $\mathbf q$ is the function to be found:
 * $\mathbf q : \left[{a \,.\,.\, b}\right] \subset \R \to X : t \mapsto x = \mathbf q \left({t}\right)$

such that:
 * $\mathbf q$ is differentiable
 * $\mathbf q \left({a}\right) = \mathbf x_a$
 * $\mathbf q \left({b}\right) = \mathbf x_b$
 * $\mathbf q'$ is the derivative of $\mathbf q$:
 * $\mathbf q' : \left[{a \,.\,.\, b}\right] \to T_{\mathbf q \left({t}\right)} X: t \mapsto v = \mathbf q' \left({t}\right)$


 * $T_{\mathbf q \left({t}\right)} X$ denotes the tangent space to $X$ at the point $\mathbf q \left({t}\right)$


 * $L$ is a real-valued function with continuous first partial derivatives:
 * $L: \left[{a \,.\,.\, b}\right] \times T X \to \R: \left({t, x, v}\right) \mapsto L \left({t, x, v}\right)$

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

The Euler–Lagrange equation, then, is given by:
 * $\displaystyle L_x \left({t, \mathbf q \left({t}\right), \mathbf q' \left({t}\right)}\right) - \dfrac {\mathrm d} {\mathrm d t} L_v \left({t, \mathbf q \left({t}\right), \mathbf q' \left({t}\right)}\right) = 0$

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.