DOC.

255

SEPTEMBER

1916 247

255. To

Constantin Carathéodory

[Berlin,]

6

September 1916

Dear

Colleague,[1]

You held out

the

prospect

of

writing

an

intuitive derivation for

me

of

the

Hamilton-Jacobi

relation.

Well,

I

succeeded in

doing so

myself

and

am showing

you my simple

considerations

only

to

spare you

the

effort. For

the

Lagrange

function L

S{ ƒ

Ldt}

=

0...

(1)

dL

d

(ÔL\

n

01

dqv

dt

(d^J

applies. Now,

we

set

ƒ

Ldt

=

J(qu, Qv,

t,

T)...

(1a)

(2)

Here

the

Qv's

are

the initial

coordinates

to

a

specified

initial time T.

I

now

consider

a

neighboring

path

(between

the

same

times

t and

T)

to be

reached

through

a

virtual

shift

in

the

path.

Through

variation

of

(2)

you

obtain,

taking

(1a)

into

account,

v-

dL

f

v-

dL

ir^

^

x

,

v-

dJ

^

dqv

6lu

^

dQv

Qv

~

^

dq"

Öqv

+

^

dQv

6Q"

From

this,

both Jacobian

equation systems

result.

Since,

first,

dL

_

dJ

dqv

VlJ

dqv

(3)

(3a)

Second,

dJ

dL

Wv'~~dQv

For

one

and

the

same

path,

however,

dL/dQv

=

Pv

is

given

as an

initial

condition,

thus

is constant;

therefore,

the

dJ/dQv’s

are

also constant

on a

single

path. If instead

of

Qv

arbitrary

functions

av

of

these

quantities

are

introduced,

we

then

naturally

also have

dJ

dcij,

=

ßv

=

const.

(3b)