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Euler-Lagrange Differential Equation


The Euler-Lagrange differential equation is the fundamental equation of calculus of variations. It states that if J is defined by an integral of the form

 J=intf(t,y,y^.)dt,
(1)

where

 y^.=(dy)/(dt),
(2)

then J has a stationary value if the Euler-Lagrange differential equation

 (partialf)/(partialy)-d/(dt)((partialf)/(partialy^.))=0
(3)

is satisfied.

If time-derivative notation y^. is replaced instead by space-derivative notation y_x, the equation becomes

 (partialf)/(partialy)-d/(dx)(partialf)/(partialy_x)=0.
(4)

The Euler-Lagrange differential equation is implemented as EulerEquations[f, u[x], x] in the Wolfram Language package VariationalMethods` .

In many physical problems, f_x (the partial derivative of f with respect to x) turns out to be 0, in which case a manipulation of the Euler-Lagrange differential equation reduces to the greatly simplified and partially integrated form known as the Beltrami identity,

 f-y_x(partialf)/(partialy_x)=C.
(5)

For three independent variables (Arfken 1985, pp. 924-944), the equation generalizes to

 (partialf)/(partialu)-partial/(partialx)(partialf)/(partialu_x)-partial/(partialy)(partialf)/(partialu_y)-partial/(partialz)(partialf)/(partialu_z)=0.
(6)

Problems in the calculus of variations often can be solved by solution of the appropriate Euler-Lagrange equation.

To derive the Euler-Lagrange differential equation, examine

deltaJ=deltaintL(q,q^.,t)dt
(7)
=int((partialL)/(partialq)deltaq+(partialL)/(partialq^.)deltaq^.)dt
(8)
=int[(partialL)/(partialq)deltaq+(partialL)/(partialq^.)(d(deltaq))/(dt)]dt,
(9)

since deltaq^.=d(deltaq)/dt. Now, integrate the second term by parts using

u=(partialL)/(partialq^.)        dv
(10)
=d(deltaq)
(11)
du=d/(dt)((partialL)/(partialq^.))dt    v=deltaq,
(12)

so

 int(partialL)/(partialq^.)(d(deltaq))/(dt)dt=int(partialL)/(partialq^.)d(deltaq)=[(partialL)/(partialq^.)deltaq]_(t_1)^(t_2)-int_(t_1)^(t_2)(d/(dt)(partialL)/(partialq^.)dt)deltaq.
(13)

Combining (◇) and (◇) then gives

 deltaJ=[(partialL)/(partialq^.)deltaq]_(t_1)^(t_2)+int_(t_1)^(t_2)((partialL)/(partialq)-d/(dt)(partialL)/(partialq^.))deltaqdt.
(14)

But we are varying the path only, not the endpoints, so deltaq(t_1)=deltaq(t_2)=0 and (14) becomes

 deltaJ=int_(t_1)^(t_2)((partialL)/(partialq)-d/(dt)(partialL)/(partialq^.))deltaqdt.
(15)

We are finding the stationary values such that deltaJ=0. These must vanish for any small change deltaq, which gives from (15),

 (partialL)/(partialq)-d/(dt)((partialL)/(partialq^.))=0.
(16)

This is the Euler-Lagrange differential equation.

The variation in J can also be written in terms of the parameter kappa as

deltaJ=int[f(x,y+kappav,y^.+kappav^.)-f(x,y,y^.)]dt
(17)
=kappaI_1+1/2kappa^2I_2+1/6kappa^3I_3+1/(24)kappa^4I_4+...,
(18)

where

v=deltay
(19)
v^.=deltay^.
(20)

and the first, second, etc., variations are

I_1=int(vf_y+v^.f_(y^.))dt
(21)
I_2=int(v^2f_(yy)+2vv^.f_(yy^.)+v^.^2f_(y^.y^.))dt
(22)
I_3=int(v^3f_(yyy)+3v^2v^.f_(yyy^.)+3vv^.^2f_(yy^.y^.)+v^.^3f_(y^.y^.y^.))dt
(23)
I_4=int(v^4f_(yyyy)+4v^3v^.f_(yyyy^.)+6v^2v^.^2f_(yyy^.y^.)+4vv^.^3f_(yy^.y^.y^.)+v^.^4f_(y^.y^.y^.y^.))dt.
(24)

The second variation can be re-expressed using

 d/(dt)(v^2lambda)=v^2lambda^.+2vv^.lambda,
(25)

so

 I_2+[v^2lambda]_2^1=int_1^2[v^2(f_(yy)+lambda^.)+2vv^.(f_(yy^.)+lambda)+v^.^2f_(y^.y^.)]dt.
(26)

But

 [v^2lambda]_2^1=0.
(27)

Now choose lambda such that

 f_(y^.y^.)(f_(yy)+lambda^.)=(f_(yy^.)+lambda)^2
(28)

and z such that

 f_(yy^.)+lambda=-(f_(yy^.))/z(dz)/(dt)
(29)

so that z satisfies

 f_(y^.y^.)z^..+f^._(y^.y^.)z^.-(f_(yy)-f^._(yy^.))z=0.
(30)

It then follows that

I_2=intf_(y^.y^.)(v^.+(f_(yy^.)+lambda)/(f_(y^.y^.))v)^2dt
(31)
=intf_(y^.y^.)(v^.-v/z(dz)/(dt))^2dt.
(32)

See also

Beltrami Identity, Brachistochrone Problem, Calculus of Variations, Euler-Lagrange Derivative, Functional Derivative, Variation

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References

Arfken, G. Mathematical Methods for Physicists, 3rd ed. Orlando, FL: Academic Press, 1985.Forsyth, A. R. Calculus of Variations. New York: Dover, pp. 17-20 and 29, 1960.Goldstein, H. Classical Mechanics, 2nd ed. Reading, MA: Addison-Wesley, p. 44, 1980.Lanczos, C. The Variational Principles of Mechanics, 4th ed. New York: Dover, pp. 53 and 61, 1986.Morse, P. M. and Feshbach, H. "The Variational Integral and the Euler Equations." §3.1 in Methods of Theoretical Physics, Part I. New York: McGraw-Hill, pp. 276-280, 1953.

Referenced on Wolfram|Alpha

Euler-Lagrange Differential Equation

Cite this as:

Weisstein, Eric W. "Euler-Lagrange Differential Equation." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/Euler-LagrangeDifferentialEquation.html

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