Dynamic Systems:

Complex Number and Harmonic Motion

By Prof. Seungchul Lee
http://iai.postech.ac.kr/
Industrial AI Lab at POSTECH

# 1. Complex NumberÂ¶

$$z_1 = a_1 + b_1i, \quad \vec{z}_1 = \begin{bmatrix} a_1 \\ b_1 \end{bmatrix}$$

$$z_2 = a_2 + b_2i, \quad \vec{z}_2 = \begin{bmatrix} a_2 \\ b_2 \end{bmatrix}$$

## 1.1. OperationÂ¶

\begin{align*} z &= z_1 + z_2 = (a_1 + a_2) + (b_1 + b_2)i \\ \\ \vec{z} &= \vec{z}_1 + \vec{z}_2 = \begin{bmatrix} a_1 \\ b_1 \end{bmatrix}+ \begin{bmatrix} a_2 \\ b_2 \end{bmatrix} = \begin{bmatrix} a_1 + a_2 \\ b_1 + b_2 \end{bmatrix} \end{align*}

• Multiply
$$\begin{cases} z_1=r_1e^{i\theta_1}\\ \\ z_2=r_2e^{i\theta_2} \end{cases} \quad\Longrightarrow\quad \begin{cases} z_1\centerdot z_2=r_1r_2e^{i(\theta_1+\theta_2)} \\ \\ {z_1 \over z_2}={r_1 \over r_2}e^{i(\theta_1-\theta_2)} \end{cases}$$

• Euler's Formula:
$$e^{i\theta} = \text{cos}\theta + i\text{sin}{\theta}$$
• Complex number in complex exponential
\begin{align*} \vec{z} &= r \, \text{cos}\theta + i\,r\,\text{sin}\theta \\ &= r\,(\text{cos}\theta + i\text{sin}\theta) \\ &= re^{i\theta}\\ \\ r &: \text{magnitude (length)} \\ \theta &: \text{phase (angle)} \end{align*}

## 1.2. Geometrical Meaning of $e^{i\theta}$Â¶

• $e^{i\theta}: \text{point on the unit circle with angle of } \theta$

• $\theta = \omega t$
• $e^{i\omega t}: \text{rotating on an unit circle with angular velocity of } \omega$.

• Question: what is the physical meaning of $e^{-i\omega t}$?

## 1.3. Sinusoidal Functions from Circular MotionsÂ¶

• Real part ($\cos$ term) is the projection onto the $Re${} axis.
• Imaginary part ($\sin$ term) is the projection onto the $Im${} axis.