446

DOC.

47 JACOBI'S THEOREM

movements

of all

points

due to

(1)

and

(2).

We call the

essence

of all these motions

a

"field of

flow"

(Strömungsfeld).

Instead of

describing

the

field

of

flow in the

sense

of

(1)

and

(2) by giving

coordinates and momenta of each

point

of

the

system as

functions

of

time,

I

can

also

think of the state

of motion-as

measured

by

the

pi-to

be

given

at

every point (qi)

as a

function

of

time,

such that the

qi

and

t

are

viewed

as

independent

variables.

Both kinds

of

description correspond exactly

to

those in

hydrodynamics

based

on

the

LAGRANGEan

or

EULERian

equations

of motions for

fluids,

respectively.

In

terms

of the second kind of

description

I have to

replace

the left-hand side

of

(1) by

dp,

dt

E

dp,

dq

dqv

dt

or, according

to

(2),

by

dp,

^

y

dH

dp,

dt

V

dp,

dqv

Therefore,

we

have,

according

to

(1),

the

system

of

equations

dPi

dt

dq.

+

*L

+

v

dH

dp,

= V

dpv dqv

'

'

(6)

The

dH-dqi

and the

dH-dpi

are

known functions

of

the

qi,

the

pi,

and

t.

Consequently,

(6)

is the

system

of

partial

differential

equations

with

which

the

components

pi

of

the

momentum vector

of the field

of

flow

comply.

The

question suggests

itself

if

there

are

fields

of

flow in which the

momentum

vector has

a

potential

such

as

to

satisfy

the conditions

dPj dpk

dqk

dq,

=

0

(7)

Pi

=

dJ

dq,'

(7a)

[p.

608]

If

(7)

is

satisfied, (6)

takes the form

dp,

dH

+

;

dt

dq,

E

dH

dpv

dpv

dq,t

=

0.

The second term is the

complete

derivation

of H

with

respect

to the coordinate

qi.

If

H denotes that function

of

qi

and time t which results from H when the

pi

in

H

are expressed

in terms

of

qi

and

t,

one gets