DOC.
1
MANUSCRIPT ON SPECIAL RELATIVITY 47
§13. Equations
of Motion of the
Material
Point
We have
already
satisfied ourselves that the
equations
of motion
of
classical
mechanics
are
not
compatible
with the
theory
of
relativity.[73]
This
presents us
with
the task
of
setting up equations
of motion for the material
point
that will
satisfy
the
requirements
of
the
theory
of
relativity.
In order
to
obtain these
equations, we
inquire
into the law of motion
of
an
electrically charged mass point
in
an
electromagnetic
field.
For the sake of
clarity,
we
confine ourselves
to
the
case
where the
accelerating
force arises
from
an
electrostatic field that is
parallel
to
the X-axis of the coordinate
system
and
where the
point
moves
along
the
X-axis.
What is the acceleration
of
the
material
point
at
an arbitrarily
chosen
space-time point
of the motion?
If
we
denote the reference
system
to
which
we
refer the
process by
E and the
instantaneous
velocity
of
the
material
point by
q,
then the
point possesses
the
velocity
q'
=
0 with
respect
to
a
reference
system
E'
that
moves
along
the
X-axis
of
the
system
E with the
velocity
v
=
q
relative
to
that
system.
But Newton's laws of
motion
are undoubtedly
valid for
infinitely
slow motions. For that
reason,
for the
immediately following
moment
of time
the motion
is
determined
by
the
equation
dq'
m
-=
etx.x
dt'
We
only
have
to transform this
equation,
which holds for
q'
=
0,
to
the
system
E in
order
to
obtain the
equation
of motion that
we are seeking,
where
m
is to
be viewed
as a
characteristic
constant
of the
mass
point
that is
not
transformed.
According
to
the
analysis
in
the
preceding
§,
e
also retains
its
value under the transformation. We
have,
further, by
virtue of the first of
equations
(23)
and because
we
have
to
set
v
=
q,
dq'
dq
\
-12
c2
2
Furthermore,
according
to
the inverse of the fourth of
equations
(IIb),
we
get,
if
we
introduce
v
=
q
and
x
=
0,
into the differentiated
equation,
dt'
+
-
dx'
dt
=
c-
=
dt'
.
N
i-ii
c2
\
i-ii
c2
Finally, according
to
the first of
equations
(21),
e'x
=
ex
Substituting
the
unprimed quantities
for the
primed
ones
in the
equation
of
motion,
[p.
38]