Oscillations - Part 2

How does $$e^{i \theta}$$ lead to oscillations?

Euler's Formula

\begin{equation} e^{i\theta} = \cos \theta + i \sin \theta \label{eq:eulersformula} \end{equation}

A number: $$z = a + b y$$ on the complex plane.

This can also be expressed in polar coordinates: $$z = r \left( \cos{\theta} + i \sin{\theta}\right)$$

Any complex number can be expressed using the exponential: $$z = |r| e^{ix}$$

Now, we have $$e^{i x} = r \left( \cos{\theta} + i \sin{\theta}\right)$$ We need to figre out what $r$ and $x$ equal.

Underdamped, revisited

The damped eq. of motion was \begin{equation} \ddot{x} + 2 \beta \dot{x} + w_0^2 x = 0 \label{eq:dampedoscillatoreqofmotion} \end{equation} which was solved by: $$x \propto e^{\alpha t}$$ where \begin{equation} \alpha = -\beta \pm \sqrt{\beta^2 - \omega_0^2} \end{equation}

If $$\beta < \omega_0$$ then we see that $\sqrt{\beta^2 - \omega^2}$ will be imaginary.

Applications

What is this?