Problems in Polar Coordinates
F. G. Aliev, Antonio Lara · 2023
One of the simplest ways of breaking the symmetry is the presence of a symmetry point in a bi-dimensional space or of a symmetry axis in three-dimensional space. The corresponding problems will be solved using cylindrical coordinates, which implies using new variables (angle, radius), keeping only a Cartesian variable to describe the space along the symmetry axis. As a consequence, the form of the Laplacian operator will change and the solutions of the Sturm-Liouville problem in the radial variable will be Bessel and Neumann functions. Also, the way to describe some features such as points, circles or thin cylinder changes, using the Dirac’s Delta function in cylindrical coordinates. This chapter is limited to problems in cylindrical coordinates in two dimensions (i.e polar coordinates), or in three when the problem is infinite along the cylinder axis.