Euler-Lagrange differential equation (advanced)

Euler-Lagrange differential equation (advanced)

Let

M

and

N

be

C

2

manifolds

. Let

L

:

M

×

T

⁢

N

→

ℝ

be twice

differentiable

. Define a functional

F

:

D

⊂

C

2

⁢

(

M

,

N

)

→

ℝ

as

F

⁢

(

q

)

=

∫

M

L

⁢

(

x

,

q

⁢

(

x

)

,

𝐃

⁢

q

⁢

(

x

)

)

⁢

d

m

⁢

x

where

D

is the subset of

http://planetmath.org/node/5555

C

2

⁢

(

M

,

N

)

for which this

integral

converges.

Note that if

f

∈

D

and

g

∈

C

2

⁢

(

M

,

N

)

and the set

{

x

∈

M

∣

f

⁢

(

x

)

≠

g

⁢

(

x

)

}

is compact, then

g

∈

D

. We may impose a topology on

D

as follows: Suppose that

f

∈

D

, that

K

⊂

M

is compact, and that

U

0

⊂

C

2

⁢

(

K

,

N

)

is open. Then we define an open set

U

⊂

D

as the set of all functions

g

∈

D

such that

f

⁢

(

x

)

=

g

⁢

(

x

)

when

x

∉

K

and such that the restriction of

g

to

K

lies in

U

0

.

It is not hard to show that the functional

F

is

continuous

in this topology, and hence it makes sense to speak of local

extrema

of

F

. Suppose that

q

0

∈

C

2

⁢

(

M

,

N

)

is a

local extremum

. Furthermore, suppose that

f

:

M

×

[

-

1

,

+

1

]

→

N

is twice differentiable and

f

⁢

(

x

,

0

)

=

q

0

⁢

(

x

)

for all

x

∈

q

0

and

f

⁢

(

x

,

y

)

=

q

0

⁢

(

x

)

for all

y

∈

[

-

1

,

+

1

]

when

x

does not lie in a certain compact subset

K

⊂

M

. Then, viewed as a map from

[

-

1

,

+

1

]

to

D

,

f

will be continuous. Therefore, since

q

0

is a local extremum of

F

,

0

wil be a local extremum of the function

y

↦

F

⁢

(

f

⁢

(

⋅

,

y

)

)

. Because the function

y

↦

F

⁢

(

f

⁢

(

⋅

,

y

)

)

is differentiable, it will be the case that

d

d

⁢

λ

⁢

F

⁢

(

f

⁢

(

⋅

,

λ

)

)

|

λ

=

0

=

0

It can be shown (see the addendum to this entry) that this condition will be satisfied if and only if

q

0

is a solution of the following

differential equation

:

d

⁢

L

-

d

⁢

(

∂

⁡

L

∂

⁡

(

d

⁢

q

)

)

=

0

.

(1)

This differential equation is known as the

Euler-Lagrange differential equation

(or Euler-Lagrange condition).

The Euler-Lagrange equation can only be used to investigate local extrema which are

smooth functions

. To a certain extent, this limitation can be ameliorated — one can study piecewise smooth functions by supplementing the Euler-Lagrange equation with auxiliary conditions at

discontinuities

and, in some cases, one can consider non-smooth solutions as weak solutions of the Euler-Lagrange equation.

In the special cases

d

⁢

L

=

0

, the Euler-Lagrange equation can be replaced by the

Beltrami identity

.

Title

Euler-Lagrange differential equation (advanced)

Canonical name

EulerLagrangeDifferentialEquationadvanced

Date of creation

2013-03-22 14:45:32

Last modified on

2013-03-22 14:45:32

Owner

rspuzio (6075)

Last modified by

rspuzio (6075)

Numerical id

16

Author

rspuzio (6075)

Entry type

Definition

Classification

msc 47A60

Synonym

Euler-Lagrange condition

Defines

Euler-Lagrange differential equation

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