Liouville’s theorem

Liouville’s theorem

Let

x

˙

=

f

⁢

(

x

)

(1)

be a

autonomous

ordinary differential equation

in

ℝ

n

defined by a smooth vector field

f

:

ℝ

n

→

ℝ

n

and the

Jacobian

of

f

is denoted

∂

⁡

f

∂

⁡

x

. Also let

Φ

t

⁢

(

x

)

be the flow (

http://planetmath.org/Flow2

) associated
with (

1

). Let

V

⁢

(

t

)

=

∫

Φ

t

⁢

(

D

)

𝑑

x

be the volume of the image of

D

under this flow after a time

t

.

Theorem 1

(Liouville’s theorem).

If

D

⊆

R

n

is a bounded

measurable

domain. Then

V

˙

⁢

(

t

)

=

∫

Φ

t

⁢

(

D

)

div

⁡

f

⁢

(

x

)

⁢

𝑑

x

Proof.

Let

V

⁢

(

t

)

be defined as above then

V

⁢

(

t

0

+

h

)

=

∫

Φ

t

0

+

h

⁢

(

D

)

𝑑

y

=

∫

Φ

h

⁢

(

Φ

t

0

⁢

(

D

)

)

𝑑

y

=

∫

Φ

t

0

⁢

(

D

)

det

⁡

(

∂

⁡

Φ

h

∂

⁡

x

⁢

(

x

)

)

⁢

𝑑

x

.

We claim that, for

x

∈

Φ

t

0

⁢

(

D

)

,

∂

⁡

Φ

t

∂

⁡

x

⁢

(

x

)

=

I

+

t

⁢

∂

⁡

f

∂

⁡

x

⁢

(

x

)

+

o

⁢

(

t

)

as

t

→

0

.

In fact,

Φ

t

⁢

(

x

)

=

x

+

∫

0

t

f

⁢

(

Φ

s

⁢

(

x

)

)

⁢

𝑑

s

,

and by the

Leibniz integral rule

∂

⁡

Φ

t

∂

⁡

x

⁢

(

x

)

=

I

+

∫

0

t

∂

∂

⁡

x

⁢

f

⁢

(

Φ

s

⁢

(

x

)

)

⁢

𝑑

s

,

so that

∂

∂

⁡

t

⁢

∂

⁡

Φ

t

∂

⁡

x

⁢

(

x

)

=

∂

∂

⁡

x

⁢

f

⁢

(

Φ

t

⁢

(

x

)

)

and evaluating at

t

=

0

we get

∂

∂

⁡

t

⁢

∂

⁡

Φ

t

∂

⁡

x

⁢

(

x

)

|

t

=

0

=

∂

∂

⁡

x

⁢

f

⁢

(

Φ

0

⁢

(

x

)

)

=

∂

⁡

f

∂

⁡

x

⁢

(

x

)

.

Our claim follows from this and from the definition of

derivative

.

Hence

det

⁡

(

∂

⁡

Φ

t

∂

⁡

x

⁢

(

x

)

)

=

det

⁡

(

I

+

t

⁢

∂

⁡

f

∂

⁡

x

⁢

(

x

)

)

+

o

⁢

(

t

)

=

∏

i

=

1

n

(

1

+

∂

⁡

f

i

∂

⁡

x

i

⁢

(

x

)

)

+

o

⁢

(

t

)

=

1

+

t

⁢

∑

i

=

1

n

∂

⁡

f

i

∂

⁡

x

i

⁢

(

x

)

+

o

⁢

(

t

)

=

1

+

t

⁢

div

⁡

f

⁢

(

x

)

+

o

⁢

(

t

)

as

t

→

0

.
It follows that

V

⁢

(

t

0

+

h

)

=

∫

Φ

t

0

⁢

(

D

)

1

+

h

⁢

div

⁡

f

⁢

(

x

)

+

o

⁢

(

h

)

⁢

d

⁢

x

and

V

˙

⁢

(

t

0

)

=

lim

h

→

0

⁡

V

⁢

(

t

0

+

h

)

-

V

⁢

(

t

0

)

h

=

∫

Φ

t

0

⁢

(

D

)

1

+

h

⁢

div

⁡

f

⁢

(

x

)

+

o

⁢

(

h

)

⁢

d

⁢

x

-

V

⁢

(

t

0

)

h

=

V

⁢

(

t

0

)

+

h

⁢

∫

Φ

t

0

⁢

(

D

)

div

⁡

f

⁢

(

x

)

⁢

𝑑

x

+

o

⁢

(

h

)

-

V

⁢

(

t

0

)

h

=

∫

Φ

t

0

⁢

(

D

)

div

⁡

f

⁢

(

x

)

⁢

𝑑

x

+

lim

h

→

0

⁡

o

⁢

(

h

)

h

=

∫

Φ

t

0

⁢

(

D

)

div

⁡

f

⁢

(

x

)

⁢

𝑑

x

.

∎

Corollary 1

.

The flow of an

Hamiltonian system

(

http://planetmath.org/HamiltonianEquations

) preserves volume.

Proof.

It follows directly since the vector field of an Hamiltonian system has

divergence

equal to zero. Hence

V

˙

=

0

implies that the volume is constant.
∎

References

TG

Teschl, Gerald: Ordinary Differential Equations and

Dynamical Systems

.

http://www.mat.univie.ac.at/ gerald/ftp/book-ode/index.html

http://www.mat.univie.ac.at/ gerald/ftp/book-ode/index.html, 2004.

Title

Liouville’s theorem

Canonical name

LiouvillesTheorem

Date of creation

2013-03-22 15:14:55

Last modified on

2013-03-22 15:14:55

Owner

Koro (127)

Last modified by

Koro (127)

Numerical id

20

Author

Koro (127)

Entry type

Theorem

Classification

msc 34A34

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