DOC. 45
QUANTUM
THEOREM
439
d
dai
Hqk,
dJ
dqk,
dJ
dt,
and these
are
quantities
that vanish due
to
(5).
From this follows that
equations
(4)
are integrated by
(6), (6a),
respectively.
§4.
The pi-field
of
a
Single
Orbit. We
come now
to
a
very
essential
point
which
I
carefully
avoided
mentioning during
the
preliminary
sketch
of
the basic idea in
§2.
In
our
deliberations in
§3
we
imagined
the
pi-field
to
be
generated by
(l
-
1)-fold
infinitely many, mutually independent
movements
that
were
illustrated
by exactly as
many
orbital
curves
in
qi-space.
But
now we
imagine
that
we
follow the undisturbed
movement
of
a single system
for
an infinitely long
time and the associated orbital
curve
to
be
(figuratively)
drawn into the
qi-space.
Here two
cases can occur.
1.
There exists
a part
of
qi-space
such that in the
course
of
time the orbital
curve
comes arbitrarily
close to
every point
of
this
(l-dimensional)
domain.
2.
The orbital
curve
in its
entirety
fits into
a
continuum
of
fewer than l
dimensions.
A
special
case
of
this
category
is
a
movement with
an exactly
closed
orbit.
Case
1
is the
general
one;
the
cases
2
result
as specializations
from
1.
As
an
example
of
case
1
we
think
of
the movement
of
a
material
point
under the
action
of
[p.
88]
a
central
force,
described
by
two coordinates which fix the
position
of
the
point
in
the orbital
plane (e.g., polar
coordinates
r
and
O).
Case 2
occurs
when the law of
attraction
is
exactly proportional
to
1/r2
and when the deviations from
a
KEPLERian
motion-which
the
theory
of
relativity
demands-are
neglected.
The
path
of
the orbit
is then closed and its
points
constitute
a
continuum
of
one
dimension. When
seen
in
three-dimensional
space,
the central
movement
is
always
a
movement
of
type
2
because the entire orbital
curve
can
be fit into
a
continuum
of
two dimensions. In
a
three-dimensional
view,
central movement has
to
be
perceived as
a
special case
of
movement,
defined
by
a
more complicated
force law
(e.g.,
the movement studied
by
Epstein
in the
theory
of
the
Stark
effect).
The
following
consideration relates to the
general case 1.
Consider
an
element
dr
of
the
qi-space.
The orbital
curve
of
the
motion under consideration
passes infinitely
often
through
dr.
There is
a
system
(pi)
of
momentum coordinates
for
each
one
of
such
crossings.
A
priori,
two
types
of orbits
are possible, obviously
of
fundamentally
different characteristics.
Type a):
The
pi-systems
repeat
such that
only
a finite
number
of
pi-systems
belong
to
dr.
The
process
of
motion
can
be
represented
with the
pi as
single-valued