Statement#

A particle of mass $m$ moves under the influence of a central force $\mathbf{F} = f(r)\hat{\mathbf{r}}$, where $f(r)$ is some function of the distance $r = |\mathbf{r}|$ from the origin, and $\hat{\mathbf{r}} = \mathbf{r}/r$ is the unit radial vector.

Prove that the angular momentum $\mathbf{L} = \mathbf{r} \times \mathbf{p}$ is conserved (constant in time).

Required Topics#

• Newton's second law
• Central forces
• Angular momentum
• Vector cross products
• Time derivatives of vector quantities
• Conservation laws

Definitions#

Central Force: A force directed along the line connecting the particle to the origin: $\mathbf{F} = f(r)\hat{\mathbf{r}}$

Angular Momentum: $\mathbf{L} = \mathbf{r} \times \mathbf{p} = \mathbf{r} \times m\mathbf{v}$

What to Prove#

Show that $\frac{d\mathbf{L}}{dt} = \mathbf{0}$, which means angular momentum is conserved.

Key Identity#

Product rule for cross products: $\frac{d}{dt}(\mathbf{A} \times \mathbf{B}) = \frac{d\mathbf{A}}{dt} \times \mathbf{B} + \mathbf{A} \times \frac{d\mathbf{B}}{dt}$

Solution#

Solution coming soon.