Question

What is azimuthal symmetry? When to use azimuthal symmetry and and how to know whether the problem has azimuthal symmetry or not?

Answer 1

If you rotate a system on an imaginary axis and the appearance of the object remains same then you'll say the object has azimuthal symmetry around that axis. For instance imagine a CD or DVD and rotate it around its center on a vertical axis passing through the center, the appearance of the CD is same, but if you choose an axis passing through it's center but on the plane of the CD itself - after an arbitrary rotation, the appearance and orientation is not same. If the CD is at the XY plane and it's center is at origin - The CD is azimuthally symmetric around Z axis. But Not X or y axis.

A baseball bat is azimuthally symmetric around the axis passes through its length , so is a cylinder.

Below are some objects and their azimuthal symmetry axis:

If you can find an axis inside a system for which you can rotate the system and the system will still look the same, that is the axis of azimuthal symmetry.

Take a rough rock - no azimuthal symmetry ,
Take a cylinder - azimuthal symmetry is there
Take a A smooth vase - azimuthal symmetry is there
Every single thing that comes straight off a pottery wheel has azimuthal symmetry.

Physical problems that include cylinder or solenoid of some kind are under azimuthal symmetry.