Science Term: Euler-Lagrange equation

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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

[tex]v(y)=\int_{x_{1}}^{x_{2}} F(x,y,y') dx \ .[/tex]

The solution usually simplifies to a second order differential equation.

Equations

[tex]F_{y}-D_{x}F_{y'} \ = \ 0[/tex]

or

[tex]\frac{\partial F}{\partial y}
\ - \ \frac{\mathrm{d} }{\mathrm{d} x} \ \frac{\partial F}{\partial y'} \ = \ 0[/tex]

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

See Also

Images

Extended explanation

PROOF

Let us find the extrema of the functional

[tex]v(y)=\int_{x_{1}}^{x_{2}}F(x,y,y')dx \ .[/tex]

Such a functional could be arc length, for example. For the variation of v,

[tex]\delta v = \frac{\partial }{\partial a}v(y+a\Delta y)|_{a=0} \ ,[/tex]

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.

[tex]\delta v=\frac{\partial }{\partial a}v(y+a\Delta y)|_{a=0} = 0[/tex]

or

[tex]\frac{\partial }{\partial a}\int_{x_{1}}^{x_{2}} F(x,y+a\Delta y,y'+a\Delta y')dx|_{a=0}=0 \ .[/tex]

Using the chain rule of multiple variables, this simplifies to

[tex]\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 \ .[/tex]

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: [itex]F_{y}=\frac{\partial F}{\partial y}[/itex])

[tex]\int_{x_{1}}^{x_{2}} (\Delta y F_{y}+\Delta y'F_{y'})dx[/tex]

Using integration by parts on the right side with "u"=F_y' and "dv"=Δy'dx:

[tex]\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 \ .[/tex]

However, Δy(x₁)=Δy(x₂)=0. Thus the middle term is zero, so:

[tex]\int_{x_{1}}^{x_{2}}\Delta y(F_{y}-\frac{\partial }{\partial x}F_{y'})dx=0 \ .[/tex]

Applying the Fundamental Lemma of Calculus of Variations, we find

[tex]F_{y}-\frac{\partial}{\partial x}F_{y'}=0 \ .[/tex]

Or, more compactly,

[tex]F_{y}-D_{x}F_{y'} = 0 \ ,[/tex]

where D_x 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 to

y + h Δy

instead. But will give it a few days first to see if anybody responds.

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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

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Entry created Jun21-09 by Pinu7