Coordinate system

We use several distinct coordinate systems in dynamics and multivariable calculus.

A note on unit vectors: “it’s easy to think [they’re] fixed in space, this is true only for the Cartesian system”.¹ When thinking about the other systems ($n-t$, $r-\theta$), the unit vectors vary depending on the point in space.

Our systems

• Rectangular coordinates
• Normal-tangential coordinates
• Polar coordinates
• Cylindrical coordinates
• Spherical coordinates

Converting between systems

To covert rectangular, polar, and $n-t$ coordinates for displacement, velocity, and acceleration, we determine the angle the polar and $n-t$ unit vectors make with the $x-y$ plane. This is computationally intensive.

The best approach is to use transformation matrices. For any orthonormal coordinate system in any dimension, where $n_i$ is the corresponding unit vector, the general transformation matrix is:

$$\text{M}=\left[\begin{array}{ccccccc}|& & |& & \dots & & |\\ {\hat{n}}_1& & {\hat{n}}_2& & \dots & & {\hat{n}}_n\\ |& & |& & \dots & & |\end{array}\right]$$

We use matrix multiplication to convert. For example, to convert from polar to rectangular coordinates:

$${\vec{r}}_{cart}=\text{M}{\vec{r}}_{pol}$$

Note that the inverse is true. Since we discuss an orthogonal matrix, the following holds:

$${\vec{r}}_{pol}={\text{M}}^{-1}{\vec{r}}_{cart}={\text{M}}^T{\vec{r}}_{cart}$$

Footnotes

1. Prof Sarris, ECE221. ↩