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
of
Qv
arbitrary
functions
av
of
these
quantities
are
introduced,
we
then
naturally
also have
dJ
dcij,
=
ßv
=
const.
(3b)
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