DOC. 47 JACOBI'S THEOREM 447
o
dt
dqt
or,
introducing
the
potential
function
J
from
(7a)
a (dJ
dqt
dt
+
H
=
0.
These
equations
are
satisfied if
one
demands for
J
the differential
equation
dJ
dt
+
H
=
0,
which is
nothing
else than the
HAMILTONian
equation
(3).
In
conjunction
with
(7a)
it solves the
equations
(6)
of
the field
of
flow.
The
equations
(5)
are
obtained
in
the
following manner.
If J
is
a complete
integral
with the
arbitrary
constants
ai,
then
(3)
remains valid
if
ai
is
replaced by
ai + dai
in
J.
Hence,
ÖV
dtdct:
y
dH
=
0
V
dpv
dqvdat
must hold. Because
of
(2)
we can
write instead
d
+
y^Z_d_)(
dJ
v
dt
dqv
dt
d0t;1
=
0.
But
the
operator
in
parentheses
is identical with the
operator
(d-dt),
a
time derivative
in
the
sense
of
the
"LAGRANGEan"
description.
Therefore,
dJ/dai
remains constant for
one
system during
its motion
and,
consequently,
an
equation system
of the form
(5)
must
hold for the
movement
of
a
point
in the
system.