Spherical Polar co-ordinates

Figure 1:- Relationship between spherical polar coordinates and the rectangular Cartesian coordinates.

The spherical polar coordinates represent the coordinates of points on the surface of a sphere in a co-variant form. The coordinates of the point in this system is represented by the radial vector which is the distance from the origin to the point, the polar or zenith angle which is the angle the radial vector makes with respect to the z axis and the azimuth or longitudinal angle which is which is the normal polar coordinate in the x − y plane as shown below in the figure.

These co-ordinates are related to the rectangular coordinates x, y, and z through

Spherical polar coordinates in terms of Cartesian coordinates are

Unit vectors in spherical coordinate system
1. : points towards the axis that is in the direction of vector along which only coordinate changes. We can also write.
2. : Unit vector is tangent at to circle . Displacement along this circle only changes coordinate.
3. : This unit vector is also tangent at if circle under consideration is produced by rotation of along z-axis. Displacement along this circle only changes .
The general differential displacement for any particle P in spherical polar coordinate is

The unit vectors , and can be expressed in terms of , and :

The unit vectors , unlike are not constant vectors but change in direction as co-ordinates and change. At each point they constitute an orthogonal right handed co-ordinate system, that is we have

Solved Problems on Spherical Polar co-ordinates

Question 1
How to Find Kinetic energy in terms of
Solution 1
We have to find the kinetic energy in terms of that is in terms of spherical c0-ordinates. Kinetic energy in terms of Cartesian co-ordinates is

(1)

Where are derivatives of z, y and z with respect to time.

Cartesian co-ordinates x, y, z in terms of and are

Now derivatives of x, y and z w.r.t. t are

Where

means all and changes with time as the particle moves or changes its position with time.

Now calculate for

Putting this value of in equation 1 we get kinetic energy of particle or system in terms of and .
Hence

Motion in two dimensions

Let us denote the two dimensional plane as XY Plane as shown below in the figure.

Figure 2:- Motion of the particle in two dimensional plane.

The position vector of the particle is given by

as for particle in motion.

In two dimension case the xy plane corresponds to the situation where . The polar angle given in figure 1 is then for the particle and spherical polar coordinates  then reduced to circular polar coordinates .

Now position of the particle P in terms of radial distance from origin O and is given by

We already have displacement vector  in three dimensionn that is

The displacement vector in circular polar coordinates can be obtained by putting and in above equation. So , we get

where unit vectors and are given by