DOC. 47 JACOBI'S THEOREM

445

Doc. 47

A

Derivation of Jacobi's Theorem

by

A. Einstein

[p.

606]

[1]

The canonical

equations

of

dynamics

dp/dt

dH

dqi

(1)

dqi

=

dH

dt

dpi,

(2)

where

H

is,

in the most

general

case,

a

function

of

the

coordinates

qi,

the momenta

pi

and time

t,

can

be

integrated-as

is well

known-according to

HAMILTON-JACOBI by

determining

a

function

J of

the

qi

and time

t

as

a

solution

of

the

partial

differential

equation

dJ

dt

+

H

=

0.

(3)

H

is here obtained from

H

by

replacing

the

pi

in H

by

the derivatives

dj/dqi.

If J

is

a

complete integral

of these

equations

with the constants of

integration

ai,

then the

system

(1),

(2)

of

the canonical

equations

is

generally integrated by

the

equations

dJ

dqi

dJ

dai

=

Bi.

(4)

=

ßi. (5)

The

more

detailed textbooks

on

dynamics verify by

calculation that

satisfying

(3),

[2]

(4),

and

(5)

has the

consequence

of

satisfying

the canonical

equations

(1),

(2).

However,

I do not know

of

a

natural

way,

free

of

surprising

tricks of the

trade,

where

one begins

at

the canonical

equations

and arrives

at

the

Hamilton-Jacobi

system

(3), (4),

(5).

Such

a

way

is

given

in the

following.

If

I

give

for

a

distinct time

t0

the coordinates

qi0

and the

associated momenta

pi0

of

the

system,

then

its motion is determined

by

(1)

and

(2).

I

represent

this motion

[p.

607]

as

the movement

of

a point

in

the n-dimensional

space

of the

coordinates

qi.

If

I

imagine

at time

t0

the momenta

pi0

given

for all

points

(qi)

of

the coordinate

space

by

means

of

the

equations

(1)

and

(2)

of

the

corresponding

system

such that the

pi0

are

continuous functions of the

qi,

then these initial conditions determine the