Velocity Vector and Spherical Coordinates

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.  These coordinates have been discussed in some detail on another page.  It is clear that if any or all  of the three coordinates, R, q, and f change, then the particle P moves relative to the origin O.  Thus if any or all of these coordinates are changing, then the particle P has a non-zero velocity.  Although the word velocity is in common use, it is only now with our discussion of the general nature of time derivatives that we can precisely define the word velocity.  The velocity of a point P relative to a coordinate system A is the time derivative with respect to that coordinate system of a position vector from a point fixed in that coordinate system to the point P.  Thus if we want to define the velocity of point P relative to the XYZ axes (system A), and we note that the origin O is fixed in these axes, then we conclude that the desired velocity is given by:

v_P/A = d ( r_P/O )/dt|_A

The above formula would be read as "the velocity of particle P relative to coordinate system A is given by the time derivative relative to coordinate system A of the position vector of the particle P relative to the point O fixed in coordinate system A.  That's a mouthful!  Now we know that this velocity must be related to the rates of change of the three coordinates R, q, and f, but how?  This is where our investigation of time derivatives in different coordinate system pays off!
 

Recall that our earlier investigation of spherical coordinates revealed several coordinate systems.  Of particular interest to us is the x, y, z system shown in the figure.  It is useful because this x axis points directly along the desired position vector from O to P.  Furthermore the length of the position vector is denoted by R (note that R may well be changing).  Thus the position vector of interest is given by:

r_P/O = R i

The velocity vector of the particle is thus the derivative of this vector:

v_P/A = d (  R i  )/dt|_A

While the above equation is accurate, it is not very useful because the unit vector i changes wildly relative to the system A (XYZ) as the particle moves.  Thus to even attempt this derivative would be an enormous challenge.  But what about our formula?  In the xyz (C) system, the unit vector i would be considered a constant.  Thus let us convert the above derivative into one evaluated in system C.  Using our formula for the difference in derivatives associated with different observers:

v_P/A = d (  R i  )/dt|_C + w_C/A X R i 

The above formula is attractive as in system C (xyz), the unit vector i can be treated as a constant.  Thus the only changing term in the product to be differentiated is the distance R.  Thus this derivative will yield the derivative with respect to time of R in the i direction.  We are further encouraged because our earlier investigation of spherical coordinates revealed an expression for the angular velocity of system C (xyz) relative to system A (XYZ).  Thus inserting this expression for the angular velocity:

v_P/A = dR/dt i +  { dq/dt ( sin f  i +   cos f k ) - df/dt j } X R i

Noting that the xyz system is a right-handed set of orthogonal axes, the required cross products can be performed just as one normally would.  Thus the desired velocity is given by:

v_P/A = dR/dt i + R dq/dt cos f  j + R df/dt k

If you compare the above result to Equation 2/18 on page 83 of our textbook, you will see they are identical.  Thus our investigation of angular velocities and time derivatives has enabled us to perform a derivation that the author deemed too complex to be included in the text!  Impressive!  Just to see if you've got it you might want to see if you can develop an expression for the acceleration in spherical coordinates.  Note that the acceleration of a particle with respect to a given reference frame (A in this case) is defined as:

a_P/A = d ( v_P/A  )/dt|_A

The above would be read as "The acceleration of a particle P with respect to coordinate system A is given by the time derivative with respect to coordinate system A of the velocity of the particle with respect to coordinate system A.  Another mouthful.  Perhaps we should just stick to calling them velocity and acceleration vectors - but remember what they really mean!

Cool stuff!  But perhaps you are still not convinced of its true utility!  Well then take a look at Problem 7/21 on page 542 of our text.  You will see spherical coordinates come to life in a machine!  Try it and see!  We will continue to see applications of the time derivatives of vectors as we investigate three dimensional dynamics!  To continue your evolution into a higher form of life, click here to return to the fundamental formulas.
 
 Copyright (1998) by Nels Madsen.  Last Updated: February 28, 1998