Continued discussion on generalizing Newton's laws to 3 dimensions.
Last time discussed taking time derivatives of vectors, and learned that time-dependent coordinate axes can introduce extra terms into expressions for the velocity and acceleration.
Worked this out explicitly for polar coordinates on the plane. Here
radial and angle unit vectors depend on the particle's polar angle,
and this dependence contributes extra terms proportional to
to the particle's velocity. We derived these extra terms,
and then derived an expression for the acceleration in polar
coordinates. This included centripetal acceleration in the radial
direction, and Coriolis acceleration in the angular direction.
Discussed why this reproduced uniform circular motion in the constant
r limit.
We then introduced spherical coordinates in 3 dimensions. For a vector
, with projection 
onto the xy-plane, we
defined 
as the polar angle between the x-axis and
. Note that the original vector lies in the
-plane. Within that plane, we defined 
as the polar
angle between the z-axis and . We can thus reproduce the
vector as follows: start with vector 
along the
z-axis, rotate it an angle in the xz-plane (i.e. about the
y-axis) to get vector 
. Then rotate an angle
about the z-axis. The result gives back the vector .
We wrote out explicit expressions for the Cartesian components x, y,
and z in terms of the spherical coordinates 
and
. We also wrote out expressions for the Cartesian components of
the unit 
, and 
vectors. Left as a
homework exercise is the next step: calculation of velocity and
acceleration in spherical coordinates.
-- KB