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
How to Find Kinetic energy in terms of
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
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
means all and changes with time as the particle moves or changes its position with time.
Now calculate for
and add them. After adding them we get
Putting this value of in equation 1 we get kinetic energy of particle or system in terms of and .
Motion in two dimensions
Let us denote the two dimensional plane as XY Plane as shown below in the figure.
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
Reference material for further reading
1. Refer this link to determine the spherical unit vectors in terms of Cartesian coordinates
2. http://planetmath.org/unitvectorsincurvilinearcoordinates to know more about curvilinear cordinates
3. Find extra stuff at this link http://mathworld.wolfram.com/SphericalCoordinates.html
Definitely there is much much more information to be added to this topic. I’ll do it some other time.