444 DOC. 46 REVIEW OF HELMHOLTZ
Doc. 46
Review
of Hermann
von
Helmholtz, Two
Lectures
on
Goethe
Not translated for this volume.
DOC. 47 JACOBI'S THEOREM
445
Doc. 47
A
Derivation of Jacobi's Theorem
by
A. Einstein
[p.
606]
[1]
The canonical
equations
of
dynamics
dp/dt
dH
dqi
(1)
dqi
=
dH
dt
dpi,
(2)
where
H
is,
in the most
general
case,
a
function
of
the
coordinates
qi,
the momenta
pi
and time
t,
can
be
integrated-as
is well
known-according to
HAMILTON-JACOBI by
determining
a
function
J of
the
qi
and time
t
as
a
solution
of
the
partial
differential
equation
dJ
dt
+
H
=
0.
(3)
H
is here obtained from
H
by
replacing
the
pi
in H
by
the derivatives
dj/dqi.
If J
is
a
complete integral
of these
equations
with the constants of
integration
ai,
then the
system
(1),
(2)
of
the canonical
equations
is
generally integrated by
the
equations
dJ
dqi
dJ
dai
=
Bi.
(4)
=
ßi. (5)
The
more
detailed textbooks
on
dynamics verify by
calculation that
satisfying
(3),
[2]
(4),
and
(5)
has the
consequence
of
satisfying
the canonical
equations
(1),
(2).
However,
I do not know
of
a
natural
way,
free
of
surprising
tricks of the
trade,
where
one begins
at
the canonical
equations
and arrives
at
the
Hamilton-Jacobi
system
(3), (4),
(5).
Such
a
way
is
given
in the
following.
If
I
give
for
a
distinct time
t0
the coordinates
qi0
and the
associated momenta
pi0
of
the
system,
then
its motion is determined
by
(1)
and
(2).
I
represent
this motion
[p.
607]
as
the movement
of
a point
in
the n-dimensional
space
of the
coordinates
qi.
If
I
imagine
at time
t0
the momenta
pi0
given
for all
points
(qi)
of
the coordinate
space
by
means
of
the
equations
(1)
and
(2)
of
the
corresponding
system
such that the
pi0
are
continuous functions of the
qi,
then these initial conditions determine the
