# Angular Momentum

## Table of Contents

## Time derivatives in a rotating frame

Let , and be unit basis vectors in the rotating frame, with a rotational speed of about the axis . Letting be the unit vector in this frame, we have:

Then if we have a function , taking the derivative in the **rotational frame** :

Which describes the change of wrt. using the rotational frame.

## Euler's Equations (rigid body dynamics)

From Time derivatives in a rotating frame we have:

where is the **moment of inertia** tensor calculated *in the intertial frame* .

It's often more useful to change to coordinates to the *rotating frame* .

With *rotating frame* we mean that the coordinate axes are rotating about a fixed axis.

Why is this useful? In this frame the moment of inertia tensor is constant (and diagonal)!

## Moment of inertia

## Covariance and contravariance

- Describe how the quantitive description of a certain geometric or physical entities change with a
**change of basis**

### Covariance

- Components of a covector change in the
*same way*as changes in scale of the reference axes → the**covector**and the reference axes are**covariant**

### Contravariance

- Components of a contr