11.3: Angular Momentum

Angular momentum $\mathbf{L}$

The notation used for the angular momentum vector is $\mathbf{L}$ where the magnitude is designated by $\left\vert \mathbf{L}\right\vert =$ $l$. Be careful not to confuse the angular momentum vector $\mathbf{L}$ with the Lagrangian $L_{cm}.$ Note that the angular momentum for two-body rotation about the center of mass with angular velocity $\omega$ is identical when evaluated in either the laboratory or equivalent two-body representation. That is, using equations $(11.2.6)$ and $(11.2.7)$

$$\mathbf{L=m}_{1}r_{1}^{\prime 2}\mathbf{\omega +m}_{2}r_{2}^{\prime 2} \mathbf{\omega =}\mu r^{2}\mathbf{\omega }$$

The center-of-mass Lagrangian leads to the following two general properties regarding the angular momentum vector $\mathbf{L}$.

1) The motion lies entirely in a plane perpendicular to the fixed direction of the total angular momentum vector. This is because

$$\mathbf{L}\cdot \mathbf{r}=\mathbf{r}\times \mathbf{p}\cdot \mathbf{r}=0$$

that is, the radius vector is in the plane perpendicular to the total angular momentum vector. Thus, it is possible to express the Lagrangian in polar coordinates, $(r,\psi )$ rather than spherical coordinates. In polar coordinates the center-of-mass Lagrangian becomes

$$L_{cm}= \frac{1}{2}\mu \left( \dot{r}^{2}+r^{2}\dot{\psi}^{2}\right) -U(r)$$

2) If the potential is spherically symmetric, then the polar angle $\psi$ is cyclic and therefore Noether’s theorem gives that the angular momentum $\mathbf{p}_{\psi }\equiv \mathbf{L}=\mathbf{r\times p}$ is a constant of motion. That is, since $\frac{\partial L_{cm}}{\partial \psi } =0,$ then the Lagrange equations imply that

$$\mathbf{\dot{p}}_{\psi }=\frac{d}{dt}\frac{\partial L_{cm}}{\partial \mathbf{ \dot{\psi}}}=0 \label{11.23}$$

where the vectors $\mathbf{\dot{p}}_{\psi }$ and $\mathbf{\dot{\psi}}$ imply that Equation $\ref{11.23}$ refers to three independent equations corresponding to the three components of these vectors. Thus the angular momentum $\mathbf{p}_{\psi },$ conjugate to $\mathbf{\psi },$ is a constant of motion. The generalized momentum $\mathbf{p} _{\psi }$ is a first integral of the motion which equals

$$\label{11.24}\mathbf{p}_{\psi }=\frac{\partial L_{cm}}{\partial \mathbf{\dot{\psi}}}=\mu r^{2}\mathbf{\dot{\psi}}=\mathbf{\hat{p}}_{\psi }l$$

where the magnitude of the angular momentum $l$, and the direction $\mathbf{ \hat{p}}_{\psi },$ both are constants of motion.

Figure $\PageIndex{1}$: Area swept out by the radius vector in the time dt.

A simple geometric interpretation of Equation $\ref{11.24}$ is illustrated in Figure $\PageIndex{1}$. The radius vector sweeps out an area $d\mathbf{A}$ in time $dt$ where

$$d\mathbf{A}=\frac{1}{2}\mathbf{r\times v}dt$$

and the vector $\mathbf{A}$ is perpendicular to the $x-y$ plane. The rate of change of area is

$$\frac{d\mathbf{A}}{dt}=\frac{1}{2}\mathbf{r\times v}$$

But the angular momentum is

$$\mathbf{L}=\mathbf{r}\times \mathbf{p}=\mu \mathbf{r\times v}=2\mu \frac{d \mathbf{A}}{dt}$$

Thus the conservation of angular momentum implies that the areal velocity $\frac{dA}{dt}$ also is a constant of motion. This fact is called Kepler’s second law of planetary motion which he deduced in $1609$ based on Tycho Brahe’s $55$ years of observational records of the motion of Mars. Kepler’s second law implies that a planet moves fastest when closest to the sun and slowest when farthest from the sun. Note that Kepler’s second law is a statement of the conservation of angular momentum which is independent of the radial form of the central potential.