Editing Euler–Lagrange equation (section)

==Generalization to manifolds==
Let <math>M</math> be a [[smooth manifold]], and let <math>C^\infty([a,b])</math> denote the space of [[smooth functions]] <math>f\colon [a,b]\to M</math>. Then, for functionals <math>S\colon C^\infty ([a,b])\to \mathbb{R}</math> of the form
:<math>
S[f]=\int_a^b (L\circ\dot{f})(t)\,\mathrm{d} t
</math>
where <math>L\colon TM\to\mathbb{R}</math> is the Lagrangian, the statement <math>\mathrm{d} S_f=0</math> is equivalent to the statement that, for all <math>t\in [a,b]</math>, each coordinate frame [[fiber bundle|trivialization]] <math>(x^i,X^i)</math> of a neighborhood of <math>\dot{f}(t)</math> yields the following <math>\dim M</math> equations:
:<math>
\forall i:\frac{\mathrm{d}}{\mathrm{d}t}\frac{\partial L}{\partial X^i}\bigg|_{\dot{f}(t)}=\frac{\partial L}{\partial x^i}\bigg|_{\dot{f}(t)}.
</math>
Euler-Lagrange equations can also be written in a coordinate-free form as <ref>{{Cite book |last1=José |last2=Saletan |year=1998 |title=Classical Dynamics: A contemporary approach |url=https://www.cambridge.org/in/academic/subjects/physics/general-and-classical-physics/classical-dynamics-contemporary-approach,%20https://www.cambridge.org/in/academic/subjects/physics/general-and-classical-physics |access-date=2023-09-12 |publisher=Cambridge University Press |language=en |isbn=9780521636360}}</ref>

:<math>
\mathcal{L}_\Delta \theta_L=dL
</math>
where <math>\theta_L</math> is the canonical momenta [[One-form (differential geometry)|1-form]] corresponding to the Lagrangian <math>L</math>. The vector field generating time translations is denoted by <math>\Delta</math> and the [[Lie derivative]] is denoted by <math>\mathcal{L}</math>. One can use local charts <math>(q^\alpha,\dot{q}^\alpha)</math> in which <math>\theta_L=\frac{\partial L}{\partial \dot{q}^\alpha}dq^\alpha</math> and <math>\Delta:=\frac{d}{dt}=\dot{q}^\alpha\frac{\partial}{\partial q^\alpha}+\ddot{q}^\alpha\frac{\partial}{\partial \dot{q}^\alpha}</math> and use coordinate expressions for the Lie derivative to see equivalence with coordinate expressions of the Euler Lagrange equation. The coordinate free form is particularly suitable for geometrical interpretation of the Euler Lagrange equations.