DOC.
10 RESEARCH NOTES
209
{J
(i^+
•
+
•
)
ds}
=
0
x% =
~
(i£)
-
l,x
[eq. 19]
J =
J
(x^+
+
)
ds
=
0
wenn
&
+
&,
+
&
=
dx
oy
dz
woraus
die
Behauptung.
[17]For
small
a, the
transformation is
unimodular
and
yields
[eq.
17]
on
integration
(with
variables
x,
t
instead of
x, y).
[18]This
transformation also
occurs
in
Einstein 1912d
(Doc. 4),
p.
456.
[19][Eq.
18]
is the
Newtonian
equation
of motion for
a
particle not subject to
external forc-
es
but
constrained
to
move on a
surface defined
by
f
=
0,
with
f
some
scalar
field.
Since
the
motion
has
constant speed
ds/dt,
the
arc
length
s
is used
to parametrize
the
motion. Con-
strained motion
is
discussed
in
Einstein's lecture
notes
on
mechanics
(Vol. 3,
Doc.
1),
[pp.
34-38,
75-76].
[20]Einstein
shows that
[eq.
18]
is
equivalent to
the
assumption
that the
particle's
trajectory
is
a
geodesic
of the surface determined
by
the
variational
principle
öjds
=
0.
From
[eq. 19],
which
is
derived with standard variational
techniques,
it
follows
that
the
acceleration
d2x/ds2
etc.
is
orthogonal to
the
deviation
vector
(^,
T|,
Q)
between the
two paths
considered
in the
calculation
and
hence
parallel to
the
normal
to
the
surface
(df/dx, df/dy, df/dz),
which
is the
content
of
[eq.
18]
See
Einstein 1912d
(Doc. 4),
p.
458, and
Einstein and Grossmann 1913
(Doc. 13),
§§1-2,
for Einstein's
use
of variational
principles
in
his theory
of
gravitation.
For
a
closely
related
discussion of
the
connection between constrained
motion
and the
concept
of
geodesic
lines
in
the
Gaussian
theory
of
surfaces,
see
Marcel Grossmann's
notes
on
Carl Friedrich Geiser's
lectures
on
Infinitesimal
Geometry
at
the ETH in
winter
semester
1897/1898
(SzZE
Biblio-
thek, Hs. 421:15).
Einstein
registered
for
this
course
(see
Vol.
1,
Appendix
E).
See
also
the
editorial
note,
"Einstein's Research
Notes
on a
Generalized
Theory
of
Relativity,"
sec.
II,
for
a
discussion of
the
role of the
Gaussian
theory
of surfaces
in the
development
of Einstein's
thinking
on
the
problem
of
gravitation.
[p.
7]
=
0.
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[21]