Overview of Coordinate Systems
Coordinates are simply a way to describe points in space using mathematics. When tackling a difficult physics problem it is generally useful to have access to a wide variety of techniques and tools. Problems can be vastly simplified by building familiarity with different coordinate systems.
- Rectangular: Coordinates are based on a square/cubic grid.
- Normal and Tangential: Coordinates are based on the trajectory of a particle.
- Polar: Coordinates are based on a circular/spherical grid (like a radar).
Rectangular Coordinates
The most well known coordinate system is the rectangular system, based on a square/cubic grid, featuring 3 axes orthogonal to one another. These are the x, y, and z axes. The associated unit vectors are \bm{\hat{i}}, \bm{\hat{j}}, and \bm{\hat{k}} respectively.
Normal and Tangential Coordinates
Normal and tangential coordinates are based on the trajectory of a particle in motion. It is exceptionally useful for problems with complex paths of motion. The coordinate system consists of a unit vector tangential to the path of motion, \bm{\hat{e}}_t, and a unit vector that points normal to the path of motion, in the direction of the center of curvature, \bm{\hat{e}}_n. The radius of curvature, \rho, is important for defining the kinematics of a particle in motion.
Polar Coordinates
Polar coordinates are based on a circular grid, defined by the radius from an origin and an angle from a given axis. The polar coordinate system consists of two orthogonal unit vectors. \bm{\hat{e}}_r always lies in the direction of \bm{\overrightarrow{r}}. \bm{\hat{e}}_\theta lies perpendicular to \bm{\hat{e}}_r in the direction of \theta.
