Euler–Lagrange equation: Difference between revisions

[[File:Tautochrone curve.gif|thumb|The Euler–Lagrange equation was in connection with their studies of the [[tautochrone]] problem. ]]
The Euler–Lagrange equation was developed in the 1750s by Euler and Lagrange in connection with their studies of the [[tautochrone]] problem. This is the problem of determining a curve on which a weighted particle will fall to a fixed point in a fixed amount of time, independent of the starting point.

Let <math>(X,L)</math> be a [[real 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}(t), {\boldsymbol v}(t))</math> the ''[[Lagrangian mechanics#Lagrangian|Lagrangian]]'', i.e. a smooth real-valued function such that <math>{\boldsymbol q}(t) \in X,</math> and <math>{\boldsymbol v}(t)</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>

A standard example{{Citation needed|reason=this example seems like a poor one as it is not even presented as a dynamical system|date=July 2023}} is finding the real-valued function ''y''(''x'') on the interval [''a'', ''b''], such that ''y''(''a'') = ''c'' and ''y''(''b'') = ''d'', for which the [[path (topology)|path]] [[arc length|length]] along the [[Curve#length of a curve|curve]] traced by ''y'' is as short as possible.

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.