DOC.
41
HAMILTON'S PRINCIPLE 243
which
we
shall
now
derive. For this
purpose
we
carry
out
an
infinitesimal
transforma-
tion of the coordinates
by setting
x'v
=
xv
+ Axv, (10)
where the
Axv
are
arbitrarily eligible, infinitesimally
small functions of the
coordinates.
x'v are
the coordinates of the world
point
in the
new system,
the
same
point
whose coordinates
were
xv
in the
original system.
Just
as
for
the
coordinates,
there
is
a
transformation law for
any
other
quantity
u, of the
type
u'
=
u
+
Au,
where
Au
must
always
be
expressible
in
terms
of
the
Axv.
From the covariant
properties
of
the
guv one
derives
easily
for the
guv
and the
guv
the
transformation
laws:
dAx,
dAxu
Ag^v
=
g/ux
+
8va
ox_
ox
a a
(11)
Agouv
aana.
oxa dxa
(12)
AR
can
be calculated with the
help
of
(11)
and
(12),
since
R
depends only upon [8]
{5}
the
guv
and the
gouv.
Thus,
one gets
the
equation
(13)
Iv^J
ÖXv
dgauv
where
we
used the abbreviation
S;
=
2|®
8?
+
©
si-
(14)
[9]
^
dgtr
From these two
equations
we
draw two conclusions that
are important
in the
following.
We
know
R/-g
to
be
an
invariant under
arbitrary
substitutions but
not
It
is,
however,
easy
to
show that the latter
quantity
is invariant under
linear
substitutions
of the coordinates.
Consequently,
the
right-hand
side
of
(13)
must
always
vanish when all
d2Axo/dxvdxa
do.
Then it
follows
that
R*
must
satisfy
the
identity
Sov =
0.
(15)
If
we
furthermore choose the
Ax
v
such that
they
differ from
zero only
inside the
domain
considered,
but vanish in
an
infinitesimal
neighborhood
of
the
boundary,
then
[p.1115]