Spherical Coordinates and Angular Velocities

The figure shows a particle (P, green) located relative to a set of axes by three coordinates, two angles and one distance.  The axes are shown in light blue and are labeled X, Y, and Z.  The position vector of the particle relative to the origin is shown in purple.  The distance coordinate is the distance of the particle from the origin (O) and is denoted R.  The two angles are formed by dropping a projection line parallel to the Z (vertical) axis from the particle until it strikes the XY (horizontal) plane.  The projection line is shown in blue.  The point of intersection with the XY plane is labeled G.  A second line (also blue) is then drawn in the XY plane from the origin to the point G.  The angular coordinate q is measured from the X axis to this second line (OG) and is positive in the sense from the +X axis toward the +Y axis.  The angle is indicated in yellow and is labeled on the figure.  The angular coordinate f is measured from this second line (OG) to the radial line from the origin to the particle (OP) and is positive in the sense from the second line toward the Z axis.  This angle is indicated in yellow and labeled in the figure.  Clearly any point in space can be located by an appropriate choice of the distance R and the two angles q and f.  Note further that values of q between 0 and 2 p radians and f between -p/2 and +p/2 radians suffice to locate any direction in space (note that other authors may define spherical coordinates slightly differently).  In fact some thought may reveal a close similarity between spherical coordinates and the use of longitude and latitude in locating points on the earths surface.
 
 

O.K., so we understand spherical coordinates, but there only seems to be one coordinate system involved and we certainly don't see any rates or axes of rotation.  The next figure helps remedy this shortcoming.  An intermediate set of axes has been added to our figure in light blue.  The x_I axis is along the line described above as being in the XY plane (OG).  The y_I axis is also in the XY plane and is perpendicular to the x_I axis.  Thus as the x_I axis makes an angle q with the X axis, so must the y_I axis make an angle q with the Y axis.  It is also useful to note that they y_I axis is perpendicular to the right triangle OGP.  Can you convince yourself that this is true?  As the z_I axis must be perpendicular to both the x_I and the y_I axes and they are both in the XY plane, then the z_I axis must be parallel to the Z axis.  Note that the origin of this new coordinate system has been shown at G but we are primarily concerned with the directions of the axes as any rotation is independent of the translation of the origin of the coordinate system.  Denoting the x_I, y_I, z_I coordinate system as system B and the X, Y, Z system as system A, we can see that the rotation of system B relative to system A is fairly simple.  System B can be seen to be rotating about the Z axis with angle of rotation q.  This is consistent with the observation that they share z axes, thus any rotation must be about that Z (or z_I) axis.  As q measures the angle between the x axes of the two systems, then their rate of rotation must be the time derivative of that angle.  Thus we can easily write the angular velocity of system B relative to system A as:

w_B/A = dq/dt K = dq/dt k_I
 
 

Wow, we have created our first (but not last) angular velocity vector!  It is interesting to note that as the two systems share the same z axis, the angular velocity vector can be expressed using unit vectors associated with either the XYZ or the xI, yI, zI system.  This raises the interesting question of relating the unit vectors from the two coordinate systems.  The figure shows a view from down the Z (or z_I) axis onto the XY plane (note that we have shifted the B system to make its origin coincident with the origin of system A without changing the directions of any of the axes).  This view makes it simple to express the unit vectors from one coordinate system in terms of the unit vectors from the other.  You should verify the following equations (expressed in matrix form):

i_I         |  cos q       sin q         0  |     I
j_I    =   | -sin  q      cos q        0  |     J
k_I        |    0              0           1  |    K

I          |  cos q      -sin q         0  |    i_I
J    =    | sin  q        cos q        0  |    j_I 
K         |    0              0           1   |   k_I
 
 

O.K., so we have an idea of how we might construct an angular velocity vector.  It still doesn't seem to have much to do with the motion of the particle.  As you might have guessed we can address this concern by introducing another coordinate system.  This coordinate system is shown in light blue in the figure with axes denoted x, y, and z (Note that the text and most authors denote these axes as R, q, and f, however we will use x, y, z to reinforce the fact that they are just a coordinate system, no better and no worse than any other coordinate system).  The x, y, z axes are shown with origin at point P.  This choice of origin is arbitrary as the direction of the axes is independent of where we position them.  The x axis, a bit difficult to see in the figure, runs along the line from the origin O to the particle P.  Thus this axis  is along the purple line OP.  The z axis is perpendicular to the x axis and is also in the OGP plane.  As the y axis is perpendicular to both x and z, it must be perpendicular to the OGP plane, thus it must be identical to the y_I axis.  Thus the xyz axes can be seen to obtained from x_I, y_I, z_I axes by rotating through the angle f about the y axis.  Thus the angular velocity of the xyz axes relative to the B system must be in the y direction (axis of rotation) and have a magnitude given by the rate of change of the angle f (rate of rotation).  Thus (can you explain the negative sign in the following?):

w_C/B = -df/dt j_I   = -df/dt j
 
 

Wow, we have created our second angular velocity vector!  Again note the two choices for the axis of rotation.  Also note the use of the right hand rule in determining the direction.  If we close the fingers on our right hand in the positive f rotation direction, our right thumb points in the -y direction.  The figure depicts a view down the positive y (or y_I) axis.  This view is useful in developing the following relationships between the unit vectors of the two coordinate systems.  Can you combine the various unit vector relationships to express i, j, and k in terms of I, J, and K?

i         |  cos f       0         sin f    |     i_I 
j    =   |   0           1            0        |     j_I
k        | -sin f       0           cos f  |    k_I 

i_I          |  cos f       0        -sin f    |     i
j_I     =   |   0           1            0        |     j
k_I         | sin f       0           cos f   |    k 

Now we have two angular velocity vectors, so what?  Well now we can use our original equation to express the angular velocity of system C relative to system A.  As we had expressions for both relative angular velocity vectors in the x_I, y_I, z_I system, we first express the angular velocity vector of system C relative to system A in terms of those unit vectors:

w_C/A = w_C/B +  w_B/A = dq/dt k_I - df/dt j_I

While the above equation is perfectly correct, there are times when it is more convenient to express that angular velocity in terms of different unit vectors.  We can use the equations given above to express the angular velocity in terms of the x, y, z unit vectors.  Thus the angular velocity of system C relative to system A can also be expressed as:
 
w_C/A = dq/dt ( sin f  i +   cos f k ) - df/dt j

Wow, we have our third angular velocity vector!  This tells us the spin rate and axis of rotation of the x, y, z system relative to the X, Y, Z system.   Now this may not seem like much of an accomplishment but look back at our figures.  Try to visualize the rotation of the x, y, z axes as the particle moves.  Clearly the rotation is complex.  Identifying the rate of rotation and axis of rotation by observing the rotation of the axes would be almost impossible.  However we have accomplished just such a task.  For given values of the angles and their derivatives with respect to time, the magnitude of the above vector (square root of the sum of the squares of the components) is the rate of rotation of the x, y, z axes relative to the fixed X, Y, Z axes.  The axis of rotation is just the direction of the angular velocity vector.  Note that while we have expressed this direction relative to the x, y, z axes, we could equally well express it in terms of any of the axis systems.  Can you express the angular velocity vector in terms of I, J and K?

O.K., so it is pretty neat that we can determine an expression for the angular velocity vector of the x, y, z system relative to the X, Y, Z system, but what good is it?  To answer that question we need to investigate the velocity and acceleration of point P as R, q and f change.  Thus we need to be able to differentiate vectors.  Thus we must return to the second of the five fundamental results!
 
Copyright (1998) by Nels Madsen.  Last Updated: February 28, 1998