Euler-Lagrange differential equation (elementary)

Euler-Lagrange differential equation (elementary)

Let

q

⁢

(

t

)

be a twice

differentiable function

from

ℝ

to

ℝ

and let

L

be a twice differentiable function from

ℝ

3

to

ℝ

. Let

q

˙

denote

d

d

⁢

t

⁢

q

.

Define the functional

I

as follows:

I

⁢

(

q

)

=

∫

a

b

L

⁢

(

t

,

q

⁢

(

t

)

,

q

˙

⁢

(

t

)

)

⁢

𝑑

t

Suppose we regard the function

L

and the limits of integratiuon

a

and

b

as fixed and allow

q

to vary. Then we could ask for which functions

q

(if any) this

integral

attains an

extremal

(

minimum

or

maximum

) value. (Note: especially in Physics literature, the function

L

is known as the

Lagrangian

.)

Suppose that a differentiable function

q

0

:

[

a

,

b

]

→

ℝ

is an

extremum

of

I

. Then, for every differentiable function

f

:

[

-

1

,

+

1

]

×

[

a

,

b

]

→

ℝ

such that

f

⁢

(

0

,

x

)

=

q

0

⁢

(

x

)

, the function

g

:

[

-

1

,

+

1

]

→

ℝ

,  defined as

g

⁢

(

λ

)

=

∫

a

b

L

⁢

(

t

,

f

⁢

(

λ

,

t

)

,

∂

⁡

f

∂

⁡

t

⁢

(

λ

,

t

)

)

⁢

𝑑

t

will have an extremum at

λ

=

0

. If this function is

differentiable

, then

d

⁢

g

/

d

⁢

λ

=

0

when

λ

=

0

.

By studying the condition

d

⁢

g

/

d

⁢

λ

=

0

(see the addendum to this entry for details), one sees that, if a function

q

is to be an extremum of the integral

I

, then

q

must satisfy the following equation:

∂

∂

⁡

q

⁢

L

-

d

d

⁢

t

⁢

(

∂

∂

⁡

q

˙

⁢

L

)

=

0

.

(1)

This equation is known as the

Euler–Lagrange

differential equation

or the Euler-Lagrange condition. A few comments on notation might be in . The notations

∂

∂

⁡

q

⁢

L

and

∂

∂

⁡

q

˙

⁢

L

denote the

partial derivatives

of the function

L

with respect to its second and third

arguments

, respectively. The notation

d

d

⁢

t

means that one is to first make the argument a function of

t

by replacing the second argument with

q

⁢

(

t

)

and the third argument with

q

˙

⁢

(

t

)

and secondly,

differentiate

the resulting function with respect to

t

. Using the

chain rule

, the Euler-Lagrange equation can be written as follows:

∂

∂

⁡

q

⁢

L

-

∂

2

∂

⁡

t

⁢

∂

⁡

q

˙

⁢

L

-

q

˙

⁢

∂

2

∂

⁡

q

⁢

∂

⁡

q

˙

⁢

L

-

q

¨

⁢

∂

2

∂

⁡

q

˙

2

⁢

L

=

0

(2)

This equation plays an important role in the

calculus of variations

. In using this equation, it must be remembered that it is only a necessary condition and, hence, given a solution of this equation, one cannot to the conclusion that this solution is a

local extremum

of the functional

F

. More work is needed to determine whether the solution of the Euler-Lagrange equation is an extremum of the integral

I

or not.

In the special case

∂

∂

⁡

t

⁢

L

=

0

, the Euler-Lagrange equation can be replaced by the

Beltrami identity

.

Title

Euler-Lagrange differential equation

(elementary)

Canonical name

EulerLagrangeDifferentialEquationelementary

Date of creation

2013-03-22 12:21:49

Last modified on

2013-03-22 12:21:49

Owner

rspuzio (6075)

Last modified by

rspuzio (6075)

Numerical id

34

Author

rspuzio (6075)

Entry type

Definition

Classification

msc 47A60

Classification

msc 70H03

Classification

msc 49K05

Synonym

Euler-Lagrange condition

Related topic

CalculusOfVariations

Related topic

BeltramiIdentity

Related topic

VersionOfTheFundamentalLemmaOfCalculusOfVariations

Defines

Euler-Lagrange differential equation

Defines

Lagrangian

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