438
DOC. 45
QUANTUM
THEOREM
Zkdpi/dpk dH/dpk.
Thus,
we
get
in
place
of
(3)
dH/dqi
+ Zk
dH/dpk
dpi/dpk
=
0.
(12)
This is
a system
of
l linear differential
equations
which
must
be
satisfied
by
the
pk
as
functions of the
qk.
We ask
now
if there
are
vector
fields for which
a
potential
J*
exists, i.e.,
which
satisfy
the conditions
(10)
and
(10a).
In this
case
(12),
due to
(10),
takes the form
dH
+ Zk
dH/dpk
dpk/dqi
=
0.
This
equation says
that
H
is
independent
of the
qi.
Therefore,
fields
of
potentials
of
the desired kind do
exist,
and their
potential
J*
satisfies the
Hamilton-Jacobi
equation
(5a),
or
J
satisfies
(5), respectively.
Thus,
it has been shown that
equations
(3)
can
be
replaced by
(7a)
and
(5a), or
by
(7)
and
(5), respectively.
We still want to show that the
equation system (4)
is
satisfied
by (6a) or
(6),
even though
it is
of
no
consequence
for the
following
deliberations.
Equations
(4)
form
a system
of
total differential
equations
for the
determination
of
the
qi
as
functions
of
time,
after the
integration
of
(5a)
and
[p. 87]
expressing
the
pi
as
functions
of
the
qi
due to
(7a).
According
to the
theory
of
differential
equations
of
the first
order,
this
system
of
total differential
equations
is
equivalent
to
the
partial
differential
equation
ZdH+dO
=
0.
(13)
k
dpk dqk
dt
The latter is satisfied
by
o
dJ
if
J
is
a
complete integral
of
(5).
Because,
if
the value
of o
is inserted into the
left-
hand side
of
(13), one
obtains under due consideration
of
(7)
dH
d2J
d2J Z
k
J dj
d4kdai
dtdai
or