Angular momentum

In classical mechanics, angular momentum is the product of the particle’s mass, velocity, and the radii about the point it is rotating, measured in $N\cdot m\cdot s$. In ${\mathbb{R}}^2$, the angular momentum vector is pointing in the $\hat{k}$ direction. We liken this to a rotational equivalent of linear momentum.

About the origin $O$:

$${\text{H}}_O=\text{r}\times m\text{v}$$

And measured with respect to some point $A$, we can calculate a scalar:

$$H_A=rmv\sin (\theta )$$

Formulas

We link momentum to moments:

$$\Sigma {\text{M}}_O={\dot{\text{H}}}_O$$ $$({\text{H}}_O{)}_1+{\int }_{t_1}^{t_2}\Sigma {\text{M}}_{O}dt=({\text{H}}_O{)}_2$$

About the mass centre $G$:

$$H_G=\Sigma {\rho }_i\times m_i{r}_i$$

Conservation

Crucially an important principle is that in a closed system, angular momentum is conserved.

$$\Delta {\text{H}}_O=0$$ $$({\text{H}}_O{)}_1=({\text{H}}_O{)}_2$$ $$(rmv{)}_{O1}=(rmv{)}_{O2}$$