DOC.
1
MECHANICS LECTURE NOTES
7
cL
( dx\
-
Ira-
dt\ dt
=
X
etc.
If
the
right-hand
side can be
directly
integrated
with
respect
to
time,
if
XYZ
vanishes
or
are is
at
least
independent
of
x
..
x
..
etc.
Example.
The
force
is
everywhere
parallel
to
a given
direction.
We choose the
one
parallel
to
the direction
Z.
Then
dx
m-
=
a
dt
dy
m-
=
b,
dt
a
dy
dy
m-
=
b,
dt
From
this,
x
=
a't
+
c1
y
=
b't
+
c2
The motion
takes place on a plane, because a
dy
-
bdx
=
0, ay
-
bx
=
const.
Remark:
the above
equation
contains the
vector
(mx, my,
mz),
the
velocity
vector
multiplied by mass. It plays a role in many derivations. We call it b
= (bx, by, bz).
We
have
dbx =
Xdt
bx
=
jXdt
The
momentum
is
equal to the time integral
of
the force acting on the body
(material
point)
[p.11]
d2x
[12]
m
dt2
=
X
d2y
m~
=
Y
dt2
d2y
d
m(1 x~
-
y-^
dt2
y
dt2
-y
Y-yX