Coordinate system comparison #rkm

The different coordinate systems and bases have different strengths and weaknesses, and no single coordinate system or basis is always best. Depending on the application, several different coordinate systems or bases may be used simultaneously for different purposes.

Name	Coords	Basis	Pros	Cons
Cartesian	$x,y,z$	$\hat\imath,\hat\jmath,\hat{k}$	No singularities or discontinuities Fixed for all time Easy differentiation Independent of origin choice	No insight into dynamics
Cylindrical or polar	$R,\theta,z$ or $r,\theta$	$\hat{e}_R,\hat{e}_\theta,\hat{e}_z$ or $\hat{e}_r,\hat{e}_\theta$	Ideal for circular and helical movement Gives insight into centripetal and coriolis acceleration Good if there is a clear motion center with 2D (single-axis) rotation	Singularity when $r = 0$ Dependent on origin choice Complex differentiation
Spherical	$r,\theta,\phi$	$\hat{e}_r,\hat{e}_\theta,\hat{e}_\phi$	Ideal for rotations in 3D Good if there is a clear motion center with 3D (multi-axis) rotation	Singularity when $r = 0$ , $\phi = 0$ , or $\phi = \pi$ Dependent on origin choice Complex differentiation
Tangential/ normal	—	$\hat{e}_t,\hat{e}_n,\hat{e}_b$	Decomposes acceleration into speed and direction changes Allows computation of radius of curvature Independent of origin choice and orientation	Discontinuous when curve direction changes Not uniquely defined when $\vec{a}$ is parallel to $\vec{v}$ Complex differentiation

Warning: Coordinate systems and bases are different. #rkm‑wc

Some coordinate systems (e.g., cylindrical, spherical) also have corresponding bases. The tangential/normal basis does not have any associated coordinate system, however. We can mix and match coordinate systems and basis. For example, we may track a point's location in polar coordinates $(r,\theta)$ , but express its velocity and acceleration in a tangential/normal basis $\hat{e}_t,\hat{e}_n$ .

Movement:	circle	var-circle	ellipse	arc
	trefoil	eight	comet	pendulum
Show:
Coords:	none	Cartesian	polar
Basis:	none	Cartesian	polar	tangential/normal
Origin:	$O_1$	$O_2$
Components:

Movement:	saddle	Viviani	eight	clover
	Lissajous	deltoid	pentagram
Show:
Coords:	none	Cartesian	cylindrical	spherical
Basis:	none	Cartesian	cylindrical	spherical	tang./norm.
Origin:	$O_1$	$O_2$
Components:

TAM 212: Introductory Dynamics

Coordinate system comparison #rkm

2D Motion #rkm‑s2

3D Motion #rkm‑s3