Category:Euler-Lagrange Equation

This category contains results about Euler-Lagrange Equation.
Definitions specific to this category can be found in Definitions/Euler-Lagrange Equation.

The Euler–Lagrange equation is an equation satisfied by a mapping $\mathbf q$ of a real variable $t$ which is a stationary point of the functional:

$\forall t \in \R: \ds \map S {\mathbf q} = \int_a^b \map L {t, \map {\mathbf q} t, \map {\mathbf q'} t} \rd t$

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.