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Line 11: ==Statement== Let <math>(X,L)</math> be a ~~mechanical~~[[dynamical system]] with <math>n</math> degrees of freedom. Here <math>X</math> is the [[configuration space (physics)\|configuration space]] and <math>L=L(t,\boldsymbol q, \boldsymbol v)</math> the ''Lagrangian'', i.e. a smooth real-valued function such that <math>\boldsymbol q \in X,</math> and <math>\boldsymbol v</math> is an <math>n</math>-dimensional "vector of speed". (For those familiar with [[differential geometry]], <math>X</math> is a [[smooth manifold]], and <math>L : {\mathbb R}_t \times TX \to {\mathbb R},</math> where <math>TX</math> is the [[tangent bundle]] of <math>X).</math> Let <math>{\cal P}(a,b,\boldsymbol x_a,\boldsymbol x_b)</math> be the set of smooth paths <math>\boldsymbol q: [a,b] \to X</math> for which <math>\boldsymbol q(a) = \boldsymbol x_a</math> and <math>\boldsymbol q(b) = \boldsymbol x_b. </math> The [[action (physics)\|action functional]] <math>S : {\cal P}(a,b,\boldsymbol x_a,\boldsymbol x_b) \to \mathbb{R}</math> is defined via

Euler–Lagrange equation: Difference between revisions