---
title: Angular Motion
categories: repetition lecture
---

# Representations of the Rotation

+ rotation matrix $R$
+ rotation around a vector $\omega$ with angle $\theta$
    + $R\omega=1\omega$, i.e. $\omega$ is an eigenvector
    + one is the only real eigenvalue
+ exponential form $\exp\hat\omega\theta$
    + Rodrigues' formula

The exponential form can be found via implicit derivation of
$R(t)R^T(t)=I$ and solution of the resulting ODE.

+ $\dot R(t)R^T(t) =\hat\omega(t)$ has to be skew-symmetric

# Formulæ

**Definition**
$$\exp(\hat\omega) = I + \sum_{i=1}^\infty \frac{(\hat\omega t)^i}{i!}.$$

**Rodrigues**
$$\exp(\hat\omega) = I +
\frac{\hat\omega}{||\omega||}\sin(||\omega||} + 
\frac{\hat\omega^2}{||\omega||^2}(1-\cos(||\omega||})$$


# Angular Velocity

Derivation is easy from the exponential form.
We have
$$\mathbf{X}(t) = \exp(\hat\omega t)\mathbf{X}_0$$
so
$$\dot\mathbf{X}=\frac{\partial}{\parial t}\exp(\hat\omega t)\mathbf{X}_0 = \hat\omega\exp(\hat\omega t)\mathbf{X}_0=\hat\omega\mathbf{X}(t)$$

The angular velocity is the vector $\omega$.