DOC.
285
DECEMBER
1916 275
On
the other
hand,
however,
the
canonical differential
equations
follow
from
(7);
for from
(7)
3
r \
_
3xk
dt^Vkdá¡
32Ù
dctjdt
3
y,
_
3xk
_
d2n 3H
doij
^
3t
doijdt
dctj
follow,
and from
this[3]
3H
_
^ dy^dx^
_
3xk
3yk
doij
^
3a.j
dt
doij
3t
(k
=
1,...n).
In combination with
the
identity
0
_
y^dÿ^dx^
_
3xk dyk
fff
dt dt dt 3t
these
last
equations
can
be written:
irr
3xk
.
3yk
3H
dH
=
Y
-~dyk
-7~dxk
+
-
dt,
Y
3t dt 3t
from which
the
canonical differential
equations
can
easily
be drawn.
Hence,
this
theorem
is
valid:
The functions
(4) are
solutions
of
the
canonical
differential equations,
in
every
case
where
an
equation
T.
ykdxk
=
d£l
+
y
Ajdcxj
+
Hdt
(8)
in
which
the
dO’s
are
a
total
differential
and
the
Aj’s
are
independent of
t
is
valid.
From this
theorem,
everything
else
follows
immediately:
I.
Canonical
transformations.
The
new
variables
Ek
=
Ek(x1...xn,
y1...yn,
t)
nk
=
nk(x1...xn,
yx...yn,
t)
J
(9)
are
such
that
a
function
P(x1...xn,
y1...yn,
t)
exists for which
yiVkdÇk
~
ykdxk)
-
db
+
p(xx.
.
.
xn,
yi.
.
.
yn)
t)dt
(10)