442 DOC. 45

QUANTUM

THEOREM

imaginable. Because,

if

J*

is the value of the

potential

of

one

point

in the

rationalized

qi-space,

the other values

are

J*

+

nh where

n

is

an integer.

Supplement

in

proof.

Further

thinking

showed that the second condition for the

applicability

of

formula

(11)

in

§4

is

always automatically fulfilled, i.e.,

the theorem

holds:

If

a

movement

produces

a pi-field,

then this

necessarily

has

a potential

J*.

According

to

Jacobi's

theorem,

every

movement

can

be derived from

a complete

integral

J*

of

(5a).

At

any rate,

there exists

at

least

one

function

J*

of

the

qi

from

which the momentum coordinates

pi

of

a system

of

movements under consideration

can

be

computed

for

every point

of

an

orbital

curve by means

of

the

equations

Pi

=

dJ*

dqi

.

We

now

have to remember

that

J*

has been obtained

by means

of

a partial

differential

equation,

i.e.,

through a prescription

of

how the function

J*

is to

be

continued in the

qi-space.

Therefore,

if

we

want to know how

J*

changes

for

a

system

in the

course

of

its

movement, we

have to envision how

J*

extends

along

an

orbital

curve (and

its

neighborhood)

according

to the differential

equation.

If

the

orbit

after

a

certain

(rather long)

time returns within close

proximity

to

a point

P

through

which the orbital

curve

has

passed

before,

then

dJ*/dq

provides

the momentum

coordinate for both times

if

we

integrate

J*

continuously along

the entire

piece

of

[p. 92]

orbital

curve

between both

points.

It is

by no

means

to

be

expected

that this

continuation leads

to

a

return to

previous

values

of

the

dJ/dq*;

instead,

one

should,

in

general, anticipate finding

a

totally

different

system

of

the

pi

every

time the

configuration

of the

qi

under consideration is

approximately

attained

again

in the

course

of

movement.

Thus,

it

is

absolutely impossible

to

represent

the

pi

as

functions

of

the

qi

for

a

motion

that continues

indefinitely.

But

if

the

pi-or

resp. a

finite

number

of value

systems

of

these quantities-repeat

under

repetition

of

the

coordinate

configuration,

the

dj*

are

expressible as

functions

of

the

qi

for

an indefinitely

continuing

motion.

Therefore,

if

a

pi-field

exists

for

an

indefinitely continuing

motion,

then there also exists

an

associated

potential

J*.

Consequently we can say

the

following:

If

there exist l

integrals

for the 21

equations

of

motion in the form

Rk

(qi,pi)

=

const, (14)