Forest UCM Ch3 AngMom

Definition of Angular Momentum

The angular momentum of a single particle is defined as

$\vec \ell = \vec r \times \vec p$

Torque

If I take the derivative of angular momentum with respect to time I get

$\vec{\dot \ell} = \frac{d}{dt} \left ( \vec r \times \vec p \right )$

$= \left ( \vec \dot r \times \vec p \right ) + \left ( \vec r \times \vec \dot p \right )$

$\left ( \vec \dot r \times \vec p \right )= \left ( \frac{1}{m} \vec p \times \vec p \right ) =0$ cross product of parallel vectors

$\left ( \vec r \times \vec \dot p \right )=\left ( \vec r \times \vec F \right )= \vec T$ Definition of Torque

$\vec T =\vec{\dot \ell}$ Newton's second law for angular motion

$\mathcal T$

Single particles

Kepler's second Law

Forest_UCM_MnAM#Angular_Momentum