Forest UCM Ch3 AngMom

Definition of Angular Momentum

The angular momentum of a single particle is defined as

[math]\vec \ell = \vec r \times \vec p[/math]

Torque

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

[math]\vec{\dot \ell} = \frac{d}{dt} \left ( \vec r \times \vec p \right )[/math]

[math]= \left ( \vec \dot r \times \vec p \right ) + \left ( \vec r \times \vec \dot p \right )[/math]

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

[math] \left ( \vec r \times \vec \dot p \right )=\left ( \vec r \times \vec F \right )= \vec T[/math] Definition of Torque

[math]\vec T =\vec{\dot \ell} [/math] Newton's second law for angular motion

[math]\cal T [/math]

Single particles

Kepler's second Law

Forest_UCM_MnAM#Angular_Momentum