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 mechanical[[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>

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}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

<math display="block"> J[f] = \int_a^b L(x,f(x),f'(x))\, \mathrm{d}x\ . </math>

Let <math>g_{\varepsilon} (x) = f (x) + \varepsilon \eta (x)</math> be the result of such a perturbation <math>\varepsilon \eta (x)</math> of <math>f</math>, where <math>\varepsilon</math> is small and <math>\eta (x)</math> is a differentiable function satisfying <math>\eta (a) = \eta (b) = 0</math>. Then define

<math display="block"> J_\Phi(\varepsilon) = J[f+\varepsilon\eta] = \int_a^b L(x,g_f(x)+\varepsilon\eta(x), g_f'(x)+\varepsilon\eta'(x) ) \, \mathrm{d}x = \int_a^b L_\varepsilon\, \mathrm{d}x .</math>

We now wish to calculate the [[total derivative]] of <math> J_\varepsilonPhi</math> with respect to ''εいぷしろん''.

<math display="block"> \begin{align}\frac{\mathrm{d} J_\varepsilonPhi}{\mathrm{d} \varepsilon} &= \frac{\mathrm d}{\mathrm d\varepsilon}\int_a^b L_L(x,f(x)+\varepsilon\eta(x), \mathrm{d}f'(x = \int_a^b \frac{\partial L_)+\varepsilon}{\partial \varepsilon}eta'(x)) \, \mathrm{d}x. </math>\\

The second line follows from the fact thatWhen <math>\varepsilon x= 0</math> does not depend on, <math> \varepsilon Phi</math>, i.e.has <math>an \frac{\partial[[extremum]] x}{\partialvalue, \varepsilon}so = 0</math>.that

<math display="block"> \int_a^b \left[ \frac{\partial L}{\partial f}(x,f(x),f'(x)) - \frac{\mathrm{d}}{\mathrm{d}x} \frac{\partial L}{\partial f'}(x,f(x),f'(x)) \right] \eta(x)\,\mathrm{d}x + \left[ \eta(x) \frac{\partial L}{\partial f'}(x,f(x),f'(x)) \right]_a^b = 0 \ . </math>

<math display="block"> \int_a^b \left[ \frac{\partial L}{\partial f}(x,f(x),f'(x)) - \frac{\mathrm{d}}{\mathrm{d}x} \frac{\partial L}{\partial f'}(x,f(x),f'(x)) \right] \eta(x)\,\mathrm{d}x = 0 \, . </math>

<math display="block"> \frac{\partial L}{\partial f}(x,f(x),f'(x)) - \frac{\mathrm{d}}{\mathrm{d}x} \frac{\partial L}{\partial f'}(x,f(x),f'(x)) = 0 \, . </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.