DOC. 45
QUANTUM
THEOREM 435
and which
is
identical to the
energy
function
if it
does
not
explicitly
contain the time
[p. 83]
t1
If
J(t,
q1...q1,
a1...al)
is
a
complete
integral
of
the
Hamilton-Jacobi
partial
differential
equations
dJ/dt
+
H(qi,dJ/dqi)
=
0,
(5)
the solution
of
the canonical
equations
is
dJ/dxi
=
Bi
(6)
dj/dqi
=
pi
(7)
If
H does not
explicitly depend upon
time-as
we
shall
assume
in the
following-we
can
solve
(5)
with
J
=
J*
-
ht,
(8)
where
h
represents a
constant and J*
no longer depends explicitly upon
the time
t.
Then
(5),
(6),
and
(7) are
replaced by
the
equations
H(qidj*/dqi)
=
h
dJ*/dai
=
Bi
dJ*/dh
=
t
-
t0
dJ*/dqi
=
pi,
(5a)
(6a)
(7a)
where, however,
the first
of
the
equations
(6a)
represents only
l
-
1
equations,
and
[5] {1}
where
al
is
replaced by
the constant
h,
and
ßl by
the constant -t0.
According
to
Epstein, the coordinates
qi
are
to
be chosen such that
a
complete
integral
of
(5a)
exists in the form
of
1Because in this
case
one
has
dH/dt
=
Zi
dH/dqiqi
+
Zi
dH/dpipi
=
0.