Applications of Polar coordinates

Question

What applications exist for Polar coordinates (especially over the more better known Cartesian coordinate system)?

Both "applied" applications and applications in pure mathematics may be included for the answer.

Answer 1

Motion of a particle subject to a central force such as gravity is much better suited to polars than Cartesians.

Calculating the areas of Cardioids, lemniscates etc are also much easier when expressed as polar equations.

Answer 2

If given problem contains polar or axis-symmetric situation of geometry ( $\theta$ all around rotation possible), geometry of surface, a rotated volume, force/torque application etc., then use of Polar coordinates is not an option, but it is forced to be used as a natural option.

In planetary motion under a central force it compels you to choose polar coordinates.

Calculation work is minimized in the suitable choice like polar/cylindrical/ toroidal system of coordinates.

If you wrongly started in Cartesian way to begin solving, seeing the extra cumbersome steps needed one would quickly switch back to the Polar.

EDIT1:

Shortest lines in axi-symmetry: Clairaut's Law states how the radial line should be disposed with respect to an arc in $ \mathbb R^2$. ( Radius* sin ( angle) = constant = minimum distance to origin). Such a concise law statement could not be made in the Cartesian coordinate system.