HOME
The Info List - Euler–Lagrange Equation





In the calculus of variations, the Euler–Lagrange equation, Euler's equation,[1] or Lagrange's equation (although the latter name is ambiguous—see disambiguation page), is a second-order partial differential equation whose solutions are the functions for which a given functional is stationary. It was developed by Swiss mathematician Leonhard Euler
Leonhard Euler
and Italian-French mathematician Joseph-Louis Lagrange
Joseph-Louis Lagrange
in the 1750s. Because a differentiable functional is stationary at its local maxima and minima, the Euler–Lagrange equation is useful for solving optimization problems in which, given some functional, one seeks the function minimizing or maximizing it. This is analogous to Fermat's theorem in calculus, stating that at any point where a differentiable function attains a local extremum its derivative is zero. In Lagrangian mechanics, because of Hamilton's principle
Hamilton's principle
of stationary action, the evolution of a physical system is described by the solutions to the Euler–Lagrange equation for the action of the system. In classical mechanics, it is equivalent to Newton's laws of motion, but it has the advantage that it takes the same form in any system of generalized coordinates, and it is better suited to generalizations. In classical field theory there is an analogous equation to calculate the dynamics of a field.

Contents

1 History 2 Statement 3 Examples 4 Generalizations for several functions, several variables, and higher derivatives

4.1 Single function of single variable with higher derivatives 4.2 Several functions of single variable with single derivative 4.3 Single function of several variables with single derivative 4.4 Several functions of several variables with single derivative 4.5 Single function of two variables with higher derivatives 4.6 Several functions of several variables with higher derivatives

5 Generalization to manifolds 6 See also 7 Notes 8 References

History[edit] 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. Lagrange solved this problem in 1755 and sent the solution to Euler. Both further developed Lagrange's method and applied it to mechanics, which led to the formulation of Lagrangian mechanics. Their correspondence ultimately led to the calculus of variations, a term coined by Euler himself in 1766.[2] Statement[edit] The Euler–Lagrange equation is an equation satisfied by a function q of a real argument t, which is a stationary point of the functional

S (

q

) =

a

b

L ( t ,

q

( t ) ,

q ˙

( t ) )

d

t

displaystyle displaystyle S( boldsymbol q )=int _ a ^ b L(t, boldsymbol q (t), boldsymbol dot q (t)),mathrm d t

where:

q

displaystyle boldsymbol q

is the function to be found:

q

: [ a , b ] ⊂

R

→ X

t

↦ x =

q

( t )

displaystyle begin aligned boldsymbol q colon [a,b]subset mathbb R &to X\t&mapsto x= boldsymbol q (t)end aligned

such that

q

displaystyle boldsymbol q

is differentiable,

q

( a ) =

x

a

displaystyle boldsymbol q (a)= boldsymbol x _ a

, and

q

( b ) =

x

b

displaystyle boldsymbol q (b)= boldsymbol x _ b

;

q ˙

displaystyle boldsymbol dot q

; is the derivative of

q

displaystyle boldsymbol q

:

q ˙

: [ a , b ]

T

q ( t )

X

t

↦ v =

q ˙

( t )

displaystyle begin aligned dot q colon [a,b]&to T_ q(t) X\t&mapsto v= dot q (t)end aligned

T

q ( t )

X

displaystyle T_ q(t) X

denotes the tangent space to

X

displaystyle X

at the point

q ( t )

displaystyle q(t)

.

L

displaystyle L

is a real-valued function with continuous first partial derivatives:

L : [ a , b ] × T X

R

( t , x , v )

↦ L ( t , x , v ) .

displaystyle begin aligned Lcolon [a,b]times TX&to mathbb R \(t,x,v)&mapsto L(t,x,v).end aligned

T X

displaystyle TX

being the tangent bundle of

X

displaystyle X

defined by

T X =

x ∈ X

x

×

T

x

X

displaystyle TX=bigcup _ xin X x times T_ x X

 ;

The Euler–Lagrange equation, then, is given by

L

x

( t , q ( t ) ,

q ˙

( t ) ) −

d

d

t

L

v

( t , q ( t ) ,

q ˙

( t ) ) = 0.

displaystyle L_ x (t,q(t), dot q (t))- frac mathrm d mathrm d t L_ v (t,q(t), dot q (t))=0.

where

L

x

displaystyle L_ x

and

L

v

displaystyle L_ v

denote the partial derivatives of

L

displaystyle L

with respect to the second and third arguments, respectively. If the dimension of the space

X

displaystyle X

is greater than 1, this is a system of differential equations, one for each component:

∂ L

q

i

( t ,

q

( t ) ,

q ˙

( t ) ) −

d

d

t

∂ L

q ˙

i

( t ,

q

( t ) ,

q ˙

( t ) ) = 0

for 

i = 1 , … , n .

displaystyle frac partial L partial q_ i (t, boldsymbol q (t), boldsymbol dot q (t))- frac mathrm d mathrm d t frac partial L partial dot q _ i (t, boldsymbol q (t), boldsymbol dot q (t))=0quad text for i=1,dots ,n.

Derivation of one-dimensional Euler–Lagrange equation

The derivation of the one-dimensional Euler–Lagrange equation is one of the classic proofs in mathematics. It relies on the fundamental lemma of calculus of variations. We wish to find a function

f

displaystyle f

which satisfies the boundary conditions

f ( a ) = A

displaystyle f(a)=A

,

f ( b ) = B

displaystyle f(b)=B

, and which extremizes the functional

J =

a

b

F ( x , f ( x ) ,

f ′

( x ) )

d

x   .

displaystyle J=int _ a ^ b F(x,f(x),f'(x)),mathrm d x .

We assume that

F

displaystyle F

is twice continuously differentiable.[3] A weaker assumption can be used, but the proof becomes more difficult.[citation needed] If

f

displaystyle f

extremizes the functional subject to the boundary conditions, then any slight perturbation of

f

displaystyle f

that preserves the boundary values must either increase

J

displaystyle J

(if

f

displaystyle f

is a minimizer) or decrease

J

displaystyle J

(if

f

displaystyle f

is a maximizer). Let

g

ε

( x ) = f ( x ) + ε η ( x )

displaystyle g_ varepsilon (x)=f(x)+varepsilon eta (x)

be the result of such a perturbation

ε η ( x )

displaystyle varepsilon eta (x)

of

f

displaystyle f

, where

ε

displaystyle varepsilon

is small and

η ( x )

displaystyle eta (x)

is a differentiable function satisfying

η ( a ) = η ( b ) = 0

displaystyle eta (a)=eta (b)=0

. Then define

J

ε

=

a

b

F ( x ,

g

ε

( x ) ,

g

ε

( x ) )

d

x =

a

b

F

ε

d

x

displaystyle J_ varepsilon =int _ a ^ b F(x,g_ varepsilon (x),g_ varepsilon '(x)),mathrm d x=int _ a ^ b F_ varepsilon ,mathrm d x

where

F

ε

= F ( x ,

g

ε

( x ) ,

g

ε

( x ) )

displaystyle F_ varepsilon =F(x,,g_ varepsilon (x),,g_ varepsilon '(x))

. We now wish to calculate the total derivative of

J

ε

displaystyle J_ varepsilon

with respect to ε.

d

J

ε

d

ε

=

d

d

ε

a

b

F

ε

d

x =

a

b

d

F

ε

d

ε

d

x

displaystyle frac mathrm d J_ varepsilon mathrm d varepsilon = frac mathrm d mathrm d varepsilon int _ a ^ b F_ varepsilon ,mathrm d x=int _ a ^ b frac mathrm d F_ varepsilon mathrm d varepsilon ,mathrm d x

It follows from the total derivative that

d

F

ε

d

ε

=

d

x

d

ε

F

ε

∂ x

+

d

g

ε

d

ε

F

ε

g

ε

+

d

g

ε

d

ε

F

ε

g

ε

=

d

g

ε

d

ε

F

ε

g

ε

+

d

g

ε

d

ε

F

ε

g

ε

= η ( x )

F

ε

g

ε

+

η ′

( x )

F

ε

g

ε

  .

displaystyle begin aligned frac mathrm d F_ varepsilon mathrm d varepsilon &= frac mathrm d x mathrm d varepsilon frac partial F_ varepsilon partial x + frac mathrm d g_ varepsilon mathrm d varepsilon frac partial F_ varepsilon partial g_ varepsilon + frac mathrm d g_ varepsilon ' mathrm d varepsilon frac partial F_ varepsilon partial g_ varepsilon ' \&= frac mathrm d g_ varepsilon mathrm d varepsilon frac partial F_ varepsilon partial g_ varepsilon + frac mathrm d g'_ varepsilon mathrm d varepsilon frac partial F_ varepsilon partial g'_ varepsilon \&=eta (x) frac partial F_ varepsilon partial g_ varepsilon +eta '(x) frac partial F_ varepsilon partial g_ varepsilon ' .\end aligned

So

d

J

ε

d

ε

=

a

b

[

η ( x )

F

ε

g

ε

+

η ′

( x )

F

ε

g

ε

]

d

x   .

displaystyle frac mathrm d J_ varepsilon mathrm d varepsilon =int _ a ^ b left[eta (x) frac partial F_ varepsilon partial g_ varepsilon +eta '(x) frac partial F_ varepsilon partial g_ varepsilon ' ,right],mathrm d x .

When ε = 0 we have gε = f, Fε = F(x, f(x), f'(x)) and Jε  has an extremum value, so that

d

J

ε

d

ε

ε = 0

=

a

b

[

η ( x )

∂ F

∂ f

+

η ′

( x )

∂ F

f ′

]

d

x = 0   .

displaystyle frac mathrm d J_ varepsilon mathrm d varepsilon bigg _ varepsilon =0 =int _ a ^ b left[eta (x) frac partial F partial f +eta '(x) frac partial F partial f' ,right],mathrm d x=0 .

The next step is to use integration by parts on the second term of the integrand, yielding

a

b

[

∂ F

∂ f

d

d

x

∂ F

f ′

]

η ( x )

d

x +

[

η ( x )

∂ F

f ′

]

a

b

= 0   .

displaystyle int _ a ^ b left[ frac partial F partial f - frac mathrm d mathrm d x frac partial F partial f' right]eta (x),mathrm d x+left[eta (x) frac partial F partial f' right]_ a ^ b =0 .

Using the boundary conditions

η ( a ) = η ( b ) = 0

displaystyle eta (a)=eta (b)=0

,

a

b

[

∂ F

∂ f

d

d

x

∂ F

f ′

]

η ( x )

d

x = 0   .

displaystyle int _ a ^ b left[ frac partial F partial f - frac mathrm d mathrm d x frac partial F partial f' right]eta (x),mathrm d x=0 .

Applying the fundamental lemma of calculus of variations now yields the Euler–Lagrange equation

∂ F

∂ f

d

d

x

∂ F

f ′

= 0   .

displaystyle frac partial F partial f - frac mathrm d mathrm d x frac partial F partial f' =0 .

Alternate derivation of one-dimensional Euler–Lagrange equation

Given a functional

J =

a

b

F ( t , y ( t ) ,

y ′

( t ) )

d

t

displaystyle J=int _ a ^ b F(t,y(t),y'(t)),mathrm d t

on

C

1

( [ a , b ] )

displaystyle C^ 1 ([a,b])

with the boundary conditions

y ( a ) = A

displaystyle y(a)=A

and

y ( b ) = B

displaystyle y(b)=B

, we proceed by approximating the extremal curve by a polygonal line with

n

displaystyle n

segments and passing to the limit as the number of segments grows arbitrarily large. Divide the interval

[ a , b ]

displaystyle [a,b]

into

n

displaystyle n

equal segments with endpoints

t

0

= a ,

t

1

,

t

2

, … ,

t

n

= b

displaystyle t_ 0 =a,t_ 1 ,t_ 2 ,ldots ,t_ n =b

and let

Δ t =

t

k

t

k − 1

displaystyle Delta t=t_ k -t_ k-1

. Rather than a smooth function

y ( t )

displaystyle y(t)

we consider the polygonal line with vertices

(

t

0

,

y

0

) , … , (

t

n

,

y

n

)

displaystyle (t_ 0 ,y_ 0 ),ldots ,(t_ n ,y_ n )

, where

y

0

= A

displaystyle y_ 0 =A

and

y

n

= B

displaystyle y_ n =B

. Accordingly, our functional becomes a real function of

n − 1

displaystyle n-1

variables given by

J (

y

1

, … ,

y

n − 1

) ≈

k = 0

n − 1

F

(

t

k

,

y

k

,

y

k + 1

y

k

Δ t

)

Δ t .

displaystyle J(y_ 1 ,ldots ,y_ n-1 )approx sum _ k=0 ^ n-1 Fleft(t_ k ,y_ k , frac y_ k+1 -y_ k Delta t right)Delta t.

Extremals of this new functional defined on the discrete points

t

0

, … ,

t

n

displaystyle t_ 0 ,ldots ,t_ n

correspond to points where

∂ J (

y

1

, … ,

y

n

)

y

m

= 0.

displaystyle frac partial J(y_ 1 ,ldots ,y_ n ) partial y_ m =0.

Evaluating this partial derivative gives

∂ J

y

m

=

F

y

(

t

m

,

y

m

,

y

m + 1

y

m

Δ t

)

Δ t +

F

y ′

(

t

m − 1

,

y

m − 1

,

y

m

y

m − 1

Δ t

)

F

y ′

(

t

m

,

y

m

,

y

m + 1

y

m

Δ t

)

.

displaystyle frac partial J partial y_ m =F_ y left(t_ m ,y_ m , frac y_ m+1 -y_ m Delta t right)Delta t+F_ y' left(t_ m-1 ,y_ m-1 , frac y_ m -y_ m-1 Delta t right)-F_ y' left(t_ m ,y_ m , frac y_ m+1 -y_ m Delta t right).

Dividing the above equation by

Δ t

displaystyle Delta t

gives

∂ J

y

m

Δ t

=

F

y

(

t

m

,

y

m

,

y

m + 1

y

m

Δ t

)

1

Δ t

[

F

y ′

(

t

m

,

y

m

,

y

m + 1

y

m

Δ t

)

F

y ′

(

t

m − 1

,

y

m − 1

,

y

m

y

m − 1

Δ t

)

]

,

displaystyle frac partial J partial y_ m Delta t =F_ y left(t_ m ,y_ m , frac y_ m+1 -y_ m Delta t right)- frac 1 Delta t left[F_ y' left(t_ m ,y_ m , frac y_ m+1 -y_ m Delta t right)-F_ y' left(t_ m-1 ,y_ m-1 , frac y_ m -y_ m-1 Delta t right)right],

and taking the limit as

Δ t → 0

displaystyle Delta tto 0

of the right-hand side of this expression yields

F

y

d

d

t

F

y ′

= 0.

displaystyle F_ y - frac mathrm d mathrm d t F_ y' =0.

The left hand side of the previous equation is the functional derivative

δ J

/

δ y

displaystyle delta J/delta y

of the functional

J

displaystyle J

. A necessary condition for a differentiable functional to have an extremum on some function is that its functional derivative at that function vanishes, which is granted by the last equation.

Examples[edit] A standard example is finding the real-valued function f on the interval [a, b], such that f(a) = c and f(b) = d, for which the path length along the curve traced by f is as short as possible.

ds

=

a

b

1 +

y

2

d

x ,

displaystyle text ds =int _ a ^ b sqrt 1+y'^ 2 ,mathrm d x,

the integrand function being L(x, y, y′) = √1 + y′ ² . The partial derivatives of L are:

∂ F ( x , y ,

y ′

)

y ′

=

y ′

1 +

y

2

and

∂ F ( x , y ,

y ′

)

∂ y

= 0.

displaystyle frac partial F(x,y,y') partial y' = frac y' sqrt 1+y'^ 2 quad text and quad frac partial F(x,y,y') partial y =0.

By substituting these into the Euler–Lagrange equation, we obtain

d

d

x

y ′

( x )

1 + (

y ′

( x )

)

2

= 0

y ′

( x )

1 + (

y ′

( x )

)

2

= C =

constant

y ′

( x )

=

C

1 −

C

2

:= A

⇒ y ( x )

= A x + B

displaystyle begin aligned frac mathrm d mathrm d x frac y'(x) sqrt 1+(y'(x))^ 2 &=0\ frac y'(x) sqrt 1+(y'(x))^ 2 &=C= text constant \Rightarrow y'(x)&= frac C sqrt 1-C^ 2 :=A\Rightarrow y(x)&=Ax+Bend aligned

that is, the function must have constant first derivative, and thus its graph is a straight line. Generalizations for several functions, several variables, and higher derivatives[edit] Single function of single variable with higher derivatives[edit] The stationary values of the functional

I [ f ] =

x

0

x

1

L

( x , f ,

f ′

,

f ″

, … ,

f

( k )

)  

d

x   ;    

f ′

:=

d

f

d

x

,  

f ″

:=

d

2

f

d

x

2

,  

f

( k )

:=

d

k

f

d

x

k

displaystyle I[f]=int _ x_ 0 ^ x_ 1 mathcal L (x,f,f',f'',dots ,f^ (k) )~mathrm d x~;~~f':= cfrac mathrm d f mathrm d x ,~f'':= cfrac mathrm d ^ 2 f mathrm d x^ 2 ,~f^ (k) := cfrac mathrm d ^ k f mathrm d x^ k

can be obtained from the Euler–Lagrange equation[4]

L

∂ f

d

d

x

(

L

f ′

)

+

d

2

d

x

2

(

L

f ″

)

− ⋯ + ( − 1

)

k

d

k

d

x

k

(

L

f

( k )

)

= 0

displaystyle cfrac partial mathcal L partial f - cfrac mathrm d mathrm d x left( cfrac partial mathcal L partial f' right)+ cfrac mathrm d ^ 2 mathrm d x^ 2 left( cfrac partial mathcal L partial f'' right)-dots +(-1)^ k cfrac mathrm d ^ k mathrm d x^ k left( cfrac partial mathcal L partial f^ (k) right)=0

under fixed boundary conditions for the function itself as well as for the first

k − 1

displaystyle k-1

derivatives (i.e. for all

f

( i )

, i ∈

0 , . . . , k − 1

displaystyle f^ (i) ,iin 0,...,k-1

). The endpoint values of the highest derivative

f

( k )

displaystyle f^ (k)

remain flexible. Several functions of single variable with single derivative[edit] If the problem involves finding several functions (

f

1

,

f

2

, … ,

f

m

displaystyle f_ 1 ,f_ 2 ,dots ,f_ m

) of a single independent variable (

x

displaystyle x

) that define an extremum of the functional

I [

f

1

,

f

2

, … ,

f

m

] =

x

0

x

1

L

( x ,

f

1

,

f

2

, … ,

f

m

,

f

1

,

f

2

, … ,

f

m

)  

d

x   ;    

f

i

:=

d

f

i

d

x

displaystyle I[f_ 1 ,f_ 2 ,dots ,f_ m ]=int _ x_ 0 ^ x_ 1 mathcal L (x,f_ 1 ,f_ 2 ,dots ,f_ m ,f_ 1 ',f_ 2 ',dots ,f_ m ')~mathrm d x~;~~f_ i ':= cfrac mathrm d f_ i mathrm d x

then the corresponding Euler–Lagrange equations are[5]

L

f

i

d

d

x

(

L

f

i

)

=

0

i

displaystyle begin aligned frac partial mathcal L partial f_ i - frac mathrm d mathrm d x left( frac partial mathcal L partial f_ i ' right)=0_ i end aligned

Single function of several variables with single derivative[edit] A multi-dimensional generalization comes from considering a function on n variables. If

Ω

displaystyle Omega

is some surface, then

I [ f ] =

Ω

L

(

x

1

, … ,

x

n

, f ,

f

, 1

, … ,

f

, n

)

d

x

  ;    

f

, j

:=

∂ f

x

j

displaystyle I[f]=int _ Omega mathcal L (x_ 1 ,dots ,x_ n ,f,f_ ,1 ,dots ,f_ ,n ),mathrm d mathbf x ,!~;~~f_ ,j := cfrac partial f partial x_ j

is extremized only if f satisfies the partial differential equation

L

∂ f

j = 1

n

x

j

(

L

f

, j

)

= 0.

displaystyle frac partial mathcal L partial f -sum _ j=1 ^ n frac partial partial x_ j left( frac partial mathcal L partial f_ ,j right)=0.

When n = 2 and functional I is the energy functional, this leads to the soap-film minimal surface problem. Several functions of several variables with single derivative[edit] If there are several unknown functions to be determined and several variables such that

I [

f

1

,

f

2

, … ,

f

m

] =

Ω

L

(

x

1

, … ,

x

n

,

f

1

, … ,

f

m

,

f

1 , 1

, … ,

f

1 , n

, … ,

f

m , 1

, … ,

f

m , n

)

d

x

  ;    

f

i , j

:=

f

i

x

j

displaystyle I[f_ 1 ,f_ 2 ,dots ,f_ m ]=int _ Omega mathcal L (x_ 1 ,dots ,x_ n ,f_ 1 ,dots ,f_ m ,f_ 1,1 ,dots ,f_ 1,n ,dots ,f_ m,1 ,dots ,f_ m,n ),mathrm d mathbf x ,!~;~~f_ i,j := cfrac partial f_ i partial x_ j

the system of Euler–Lagrange equations is[4]

L

f

1

j = 1

n

x

j

(

L

f

1 , j

)

=

0

1

L

f

2

j = 1

n

x

j

(

L

f

2 , j

)

=

0

2

L

f

m

j = 1

n

x

j

(

L

f

m , j

)

=

0

m

.

displaystyle begin aligned frac partial mathcal L partial f_ 1 -sum _ j=1 ^ n frac partial partial x_ j left( frac partial mathcal L partial f_ 1,j right)&=0_ 1 \ frac partial mathcal L partial f_ 2 -sum _ j=1 ^ n frac partial partial x_ j left( frac partial mathcal L partial f_ 2,j right)&=0_ 2 \vdots qquad vdots qquad &quad vdots \ frac partial mathcal L partial f_ m -sum _ j=1 ^ n frac partial partial x_ j left( frac partial mathcal L partial f_ m,j right)&=0_ m .end aligned

Single function of two variables with higher derivatives[edit] If there is a single unknown function f to be determined that is dependent on two variables x1 and x2 and if the functional depends on higher derivatives of f up to n-th order such that

I [ f ]

=

Ω

L

(

x

1

,

x

2

, f ,

f

, 1

,

f

, 2

,

f

, 11

,

f

, 12

,

f

, 22

, … ,

f

, 22 … 2

)

d

x

f

, i

:=

∂ f

x

i

,

f

, i j

:=

2

f

x

i

x

j

,

displaystyle begin aligned I[f]&=int _ Omega mathcal L (x_ 1 ,x_ 2 ,f,f_ ,1 ,f_ ,2 ,f_ ,11 ,f_ ,12 ,f_ ,22 ,dots ,f_ ,22dots 2 ),mathrm d mathbf x \&qquad quad f_ ,i := cfrac partial f partial x_ i ;,quad f_ ,ij := cfrac partial ^ 2 f partial x_ i partial x_ j ;,;;dots end aligned

then the Euler–Lagrange equation is[4]

L

∂ f

x

1

(

L

f

, 1

)

x

2

(

L

f

, 2

)

+

2

x

1

2

(

L

f

, 11

)

+

2

x

1

x

2

(

L

f

, 12

)

+

2

x

2

2

(

L

f

, 22

)

− ⋯ + ( − 1

)

k

k

x

2

k

(

L

f

, 22 … 2

)

= 0

displaystyle begin aligned frac partial mathcal L partial f &- frac partial partial x_ 1 left( frac partial mathcal L partial f_ ,1 right)- frac partial partial x_ 2 left( frac partial mathcal L partial f_ ,2 right)+ frac partial ^ 2 partial x_ 1 ^ 2 left( frac partial mathcal L partial f_ ,11 right)+ frac partial ^ 2 partial x_ 1 partial x_ 2 left( frac partial mathcal L partial f_ ,12 right)+ frac partial ^ 2 partial x_ 2 ^ 2 left( frac partial mathcal L partial f_ ,22 right)\&-dots +(-1)^ k frac partial ^ k partial x_ 2 ^ k left( frac partial mathcal L partial f_ ,22dots 2 right)=0end aligned

which can be represented shortly as:

L

∂ f

+

j = 1

n

μ

1

≤ … ≤

μ

j

( − 1

)

j

j

x

μ

1

… ∂

x

μ

j

(

L

f

,

μ

1

μ

j

)

= 0

displaystyle frac partial mathcal L partial f +sum _ j=1 ^ n sum _ mu _ 1 leq ldots leq mu _ j (-1)^ j frac partial ^ j partial x_ mu _ 1 dots partial x_ mu _ j left( frac partial mathcal L partial f_ ,mu _ 1 dots mu _ j right)=0

wherein

μ

1

μ

j

displaystyle mu _ 1 dots mu _ j

are indices that span the number of variables, that is, here they go from 1 to 2. Here summation over the

μ

1

μ

j

displaystyle mu _ 1 dots mu _ j

indices is only over

μ

1

μ

2

≤ … ≤

μ

j

displaystyle mu _ 1 leq mu _ 2 leq ldots leq mu _ j

in order to avoid counting the same partial derivative multiple times, for example

f

, 12

=

f

, 21

displaystyle f_ ,12 =f_ ,21

appears only once in the previous equation. Several functions of several variables with higher derivatives[edit] If there are p unknown functions fi to be determined that are dependent on m variables x1 ... xm and if the functional depends on higher derivatives of the fi up to n-th order such that

I [

f

1

, … ,

f

p

]

=

Ω

L

(

x

1

, … ,

x

m

;

f

1

, … ,

f

p

;

f

1 , 1

, … ,

f

p , m

;

f

1 , 11

, … ,

f

p , m m

; … ;

f

p , m … m

)

d

x

f

i , μ

:=

f

i

x

μ

,

f

i ,

μ

1

μ

2

:=

2

f

i

x

μ

1

x

μ

2

,

displaystyle begin aligned I[f_ 1 ,ldots ,f_ p ]&=int _ Omega mathcal L (x_ 1 ,ldots ,x_ m ;f_ 1 ,ldots ,f_ p ;f_ 1,1 ,ldots ,f_ p,m ;f_ 1,11 ,ldots ,f_ p,mm ;ldots ;f_ p,mldots m ),mathrm d mathbf x \&qquad quad f_ i,mu := cfrac partial f_ i partial x_ mu ;,quad f_ i,mu _ 1 mu _ 2 := cfrac partial ^ 2 f_ i partial x_ mu _ 1 partial x_ mu _ 2 ;,;;dots end aligned

where

μ

1

μ

j

displaystyle mu _ 1 dots mu _ j

are indices that span the number of variables, that is they go from 1 to m. Then the Euler–Lagrange equation is

L

f

i

+

j = 1

n

μ

1

≤ … ≤

μ

j

( − 1

)

j

j

x

μ

1

… ∂

x

μ

j

(

L

f

i ,

μ

1

μ

j

)

= 0

displaystyle frac partial mathcal L partial f_ i +sum _ j=1 ^ n sum _ mu _ 1 leq ldots leq mu _ j (-1)^ j frac partial ^ j partial x_ mu _ 1 dots partial x_ mu _ j left( frac partial mathcal L partial f_ i,mu _ 1 dots mu _ j right)=0

where the summation over the

μ

1

μ

j

displaystyle mu _ 1 dots mu _ j

is avoiding counting the same derivative

f

i ,

μ

1

μ

2

=

f

i ,

μ

2

μ

1

displaystyle f_ i,mu _ 1 mu _ 2 =f_ i,mu _ 2 mu _ 1

several times, just as in the previous subsection. This can be expressed more compactly as

j = 0

n

μ

1

≤ … ≤

μ

j

( − 1

)

j

μ

1

μ

j

j

(

L

f

i ,

μ

1

μ

j

)

= 0

displaystyle sum _ j=0 ^ n sum _ mu _ 1 leq ldots leq mu _ j (-1)^ j partial _ mu _ 1 ldots mu _ j ^ j left( frac partial mathcal L partial f_ i,mu _ 1 dots mu _ j right)=0

Generalization to manifolds[edit] Let

M

displaystyle M

be a smooth manifold, and let

C

( [ a , b ] )

displaystyle C^ infty ([a,b])

denote the space of smooth functions

f : [ a , b ] → M

displaystyle f:[a,b]to M

. Then, for functionals

S :

C

( [ a , b ] ) →

R

displaystyle S:C^ infty ([a,b])to mathbb R

of the form

S [ f ] =

a

b

( L ∘

f ˙

) ( t )

d

t

displaystyle S[f]=int _ a ^ b (Lcirc dot f )(t),mathrm d t

where

L : T M →

R

displaystyle L:TMto mathbb R

is the Lagrangian, the statement

d

S

f

= 0

displaystyle mathrm d S_ f =0

is equivalent to the statement that, for all

t ∈ [ a , b ]

displaystyle tin [a,b]

, each coordinate frame trivialization

(

x

i

,

X

i

)

displaystyle (x^ i ,X^ i )

of a neighborhood of

f ˙

( t )

displaystyle dot f (t)

yields the following

dim ⁡ M

displaystyle dim M

equations:

∀ i :

d

d

t

∂ F

X

i

f ˙

( t )

=

∂ F

x

i

f ˙

( t )

displaystyle forall i: frac mathrm d mathrm d t frac partial F partial X^ i bigg _ dot f (t) = frac partial F partial x^ i bigg _ dot f (t)

See also[edit]

Look up Euler–Lagrange equation in Wiktionary, the free dictionary.

Lagrangian mechanics Hamiltonian mechanics Analytical mechanics Beltrami identity Functional derivative

Notes[edit]

^ Fox, Charles (1987). An introduction to the calculus of variations. Courier Dover Publications. ISBN 978-0-486-65499-7.  ^ A short biography of Lagrange Archived 2007-07-14 at the Wayback Machine. ^ Courant & Hilbert 1953, p. 184 ^ a b c Courant, R; Hilbert, D (1953). Methods of Mathematical Physics. Vol. I (First English ed.). New York: Interscience Publishers, Inc. ISBN 978-0471504474.  ^ Weinstock, R. (1952). Calculus
Calculus
of Variations with Applications to Physics and Engineering. New York: McGraw-Hill. 

References[edit]

Hazewinkel, Michiel, ed. (2001) [1994], "Lagrange equations (in mechanics)", Encyclopedia of Mathematics, Springer Science+Business Media B.V. / Kluwer Academic Publishers, ISBN 978-1-55608-010-4  Weisstein, Eric W. "Euler-Lagrange Differential Equation". MathWorld.  " Calculus
Calculus
of Variations". PlanetMath.  Gelfand, Izrail Moiseevich (1963). Calculus
Calculus
of Variations. Dover. ISBN 0-486-41448-5.  Roubicek, T.: Calculus
Calculus
of variations. Chap.17 in: Mathematical Tools for Physicists. (Ed. M. Grinfeld) J. Wiley, Weinheim, 2014, ISBN 978-3-527-41188-7,

.