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)