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Line 221: ==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)}.

Euler–Lagrange equation: Difference between revisions