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Euler-Lagrange equation

 Definition/Summary Also known as the Euler equation. It is the solution to finding an extrema of a functional in the form of $$v(y)=\int_{x_{1}}^{x_{2}} F(x,y,y') dx \ .$$ The solution usually simplifies to a second order differential equation.

 Equations $$F_{y}-D_{x}F_{y'} \ = \ 0$$ or $$\frac{\partial F}{\partial y} \ - \ \frac{\mathrm{d} }{\mathrm{d} x} \ \frac{\partial F}{\partial y'} \ = \ 0$$

 Scientists Leonhard Euler (1707-1783) Joseph-Louis, comte de Lagrange (1736-1813)

 Recent forum threads on Euler-Lagrange equation

 Breakdown Mathematics > Calculus/Analysis >> Calculus of Variations

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 Extended explanation PROOF Let us find the extrema of the functional $$v(y)=\int_{x_{1}}^{x_{2}}F(x,y,y')dx \ .$$ Such a functional could be arc length, for example. For the variation of v, $$\delta v = \frac{\partial }{\partial a}v(y+a\Delta y)|_{a=0} \ ,$$ let Δy be an arbitrary differentiable function such that Δy(x1)=Δy(x2)=0. Now, to find the extrema, the variation must be zero. i.e. $$\delta v=\frac{\partial }{\partial a}v(y+a\Delta y)|_{a=0} = 0$$ or $$\frac{\partial }{\partial a}\int_{x_{1}}^{x_{2}} F(x,y+a\Delta y,y'+a\Delta y')dx|_{a=0}=0 \ .$$ Using the chain rule of multiple variables, this simplifies to $$\int_{x_{1}}^{x_{2}} (\frac{\partial F}{\partial y}\frac{\mathrm{d} (y+a\Delta y)}{\mathrm{d} a}+\frac{\partial F}{\partial y'}\frac{\mathrm{d}(y'+a\Delta y' ) }{\mathrm{d} a}) dx \ .$$ We then split d(y+aΔy) and d(y'+aΔy') into dy+Δyda and dy'+Δy'da respectively. Remember that y and y' is independent of a, and da/da=1. We therefore get (using different notation: $F_{y}=\frac{\partial F}{\partial y}$) $$\int_{x_{1}}^{x_{2}} (\Delta y F_{y}+\Delta y'F_{y'})dx$$ Using integration by parts on the right side with "u"=Fy' and "dv"=Δy'dx: $$\int_{x_{1}}^{x_{2}}\Delta yF_{y} dx \ + \ [\Delta yF_{y'}]_{x_{1}}^{x_{2}} \ - \ \int_{x_{1}}^{x_{2}}\Delta y\frac{\mathrm{d} F_{y'}}{\mathrm{d} x} \ dx \ .$$ However, Δy(x1)=Δy(x2)=0. Thus the middle term is zero, so: $$\int_{x_{1}}^{x_{2}}\Delta y(F_{y}-\frac{\partial }{\partial x}F_{y'})dx=0 \ .$$ Applying the Fundamental Lemma of Calculus of Variations, we find $$F_{y}-\frac{\partial}{\partial x}F_{y'}=0 \ .$$ Or, more compactly, $$F_{y}-D_{x}F_{y'} = 0 \ ,$$ where Dx is the differential operator with respect to x. This is a second order differential equation which, when solved, gives the desired extrema of the functional.

Commentary

 lpetrich @ 07:52 PM Jul18-12 I think that this discussion ought to mention generalization to multiple independent variables x and multiple dependent variables y, complete with noting that the integral becomes an integral over all the x's. I'm half-thinking of starting an article on the Lagrangian and how it can be physically motivated. I'll mention some notable Lagrangians.

 tiny-tim @ 04:51 AM Jun29-09 he he I've removed the extra "n" from "Langrange" (now, why did nonbody spot that? )

 Pinu7 @ 11:02 PM Jun28-09 Oooops, I misread your former comment. Sorry, I am usually careless about my limits of integration.

 Redbelly98 @ 11:17 AM Jun28-09 I've never seen a discussion in a textbook where the same symbol is used to represent two different quantities, so I've changed the integration limits from a and b to x1 and x2. (This is what they were in the Definition/Summary section originally.) Though I've made changes to its format, this entry is still chiefly a result of your efforts. Thanks for your contribution Pinu7.

 Pinu7 @ 04:25 PM Jun27-09 Redbelly, I'm not sure. I have seen several texts on the calculus of variations, and they all used different notations including a,h, and epsilon. So I think the issue is trivial.

 Redbelly98 @ 06:54 PM Jun25-09 Just realized a problem. The symbol a refers to both the lower limit of integration, as well as the variation parameter in y+aΔy. Is there a standard notation to change one of these to? If not, I'll probably change it toy + h Δyinstead. But will give it a few days first to see if anybody responds. ===== EDITS - Changed one occurrence of v(f + a Δf) to v(y + a Δy), to be consistent with the rest of the article. - Other minor changes.

 Pinu7 @ 03:24 PM Jun23-09 Thanks, Redbelly.

 Redbelly98 @ 10:33 AM Jun23-09 June 23, 2009 - corrected typos - put equations on separate lines, rather than inline with text - minor LaTex formatting changes ===== Entry created Jun21-09 by Pinu7