6
DOC.
1
MECHANICS LECTURE NOTES
The
equations
of
motion
given
above have
the character of
definitional
equations
for
the
force,
thus
they
can
be
neither confirmed
nor
refuted
by experience.
Nonetheless,
we
could find
ourselves
compelled
by
experience
to
abandon
them; this would
happen
dbc
if the
description
of
facts
by means
of the
equations
m
-
=
X
. .
would
lead
to
our
dt2
having
to
assume
expressions
for the
force
components
X
. . .
in
a very
complicated
manner.
One
would
then
reject
the
equations
of motion
as
unsuitable.
Example:
identical
springs,
stretched
in
the
same
way,
act in
the
same
direction
upon
a
free
body.
If the acceleration
were
not
proportional
to
the number of
springs
acting,
then
it would follow from
the
equations
that the
force would also not
be
proportional
to
the number of
springs.
This does not
represent
a logical
contradiction,
but
it would
[p. 9]
result
in
our presuming
that
we
could arrive
at
a
simpler,
i.e., preferable theory
of
motion,
if
we
based
ourselves
on
other
equations
of
motion.
General Remarks
on
the Motion of
the
Material
Point
For
our
equations
of motion
to
be
useful,
the
expressions
for the
force
components
X
etc.
may
not
contain
higher
than first-order
time derivatives
of
the
coordinates.
Because
the
second derivative
can
be
eliminated
by
solving
the
equations.
However,
the
occurrence
of
higher
derivatives would
make
a
solution for the
second derivative
seem
unjustified.
Hence,
for
a
general theory
we
have to
consider
X
etc.
as
functions
of
dx
xyz-
...
and
t
alone.
We
have
then
3
simultaneous
equations
of the
second
order.
The
general
integrals
of these
eq.
contain
6 arbitrary
constants.
For the motion
is
completely
determined
only if,
for
a
time
t0,
x
.
.
and
x
y z
are given.
If X
.
.
are
unique functions,
then the solution
is
thereby uniquely
det.
For
we
can
write
d^
=
-X(x..x...t)
"4
=
* dx
=
X(
)dt
dx
=
xdt
dtm dt
------- oder----- ---
------- ----- -----
[p. 10]
Thus,
if
x
..
x
..
are given
for
a
time
t,
they
can
be calculated for
the
time t
+
dt
etc.
In
certain
cases
it
proves possible
to
integrate
the
equations
of motion
once (first
integrals), so
that
one
arrives
at
first-order
equations.
(1)
The
eq.
of
mot.
can
be written