Central forces and angular momentum

I stumbled on some half-finished notes from a few years back when I was messing around with Newton's second law in polar coordinates.

The analysis lets one recover a few important results, such as conservation of angular momentum as shown here, as well as more specific descriptions of the movement of a particle experiencing radial forces of different kinds, which I'll return to in later posts.

The starting point is the familiar form of Newton's second law, \begin{align} \vec F &= m\cdot \vec a = m\cdot \frac{\mathrm d^2\vec r}{\mathrm d t^2}\,, \end{align} and the position vector's polar coordinate, \begin{align} \vec r &= \vec r\left(r(t), \theta(t)\right) = r(t)\hat r\,. \end{align}

The coordinate system is illustrated here:

The unit vectors defining the polar coordinate system are $\hat r$ and $\hat \theta$, which depend on $t$ through the coordinates $r$ and $\theta$. Their derivatives w.r.t. $t$ are found using the chain rule: \begin{align} \frac{\mathrm d \hat r}{\mathrm d t} &= \frac{\partial \hat r}{\partial r} \frac{\mathrm d r}{\mathrm d t} + \frac{\partial \hat r}{\partial \theta} \frac{\mathrm d \theta}{\mathrm d t} \\ &= \dot\theta \hat\theta\,,\\ \frac{\mathrm d \hat \theta}{\mathrm d t} &= \frac{\partial \hat \theta}{\partial r} \frac{\mathrm d r}{\mathrm d t} + \frac{\partial \hat \theta}{\partial \theta} \frac{\mathrm d \theta}{\mathrm d t} \\ &= - \dot\theta \hat r\,. \end{align}

The first and second time derivatives of position $\vec r$ are velocity $\vec v$ and acceleration $\vec a$ respectively, which are obtained using the product rule and chain rule: \begin{align} \vec v = \frac{\mathrm d \vec r}{\mathrm d t} &= \dot r \hat r + r \frac{\mathrm d\hat r}{\mathrm d t}\\ &= \dot r\hat r + r \dot \theta \hat \theta\,,\\ \vec a = \frac{\mathrm d^2 \vec r}{\mathrm d t^2} &= \frac{\mathrm d}{\mathrm d t}\left(\dot r\hat r + r \dot \theta \hat \theta\right)\\ &= \ddot r \hat r + \dot r \frac{\mathrm d \hat r}{\mathrm d t} + \dot r \dot\theta \hat\theta + r \ddot \theta \hat \theta + r \dot \theta \frac{\mathrm d \hat\theta}{\mathrm d t}\\ &= \ddot r \hat r + \dot r\dot \theta \hat \theta + \dot r\dot \theta\hat \theta + r \ddot\theta \hat \theta - r \dot\theta^2 \hat r\\ &= \left(\ddot r - r \dot\theta^2\right)\hat r + \left(2 \dot r\dot \theta + r \ddot \theta\right)\hat \theta\,, \end{align} giving Newton's second law as \begin{align} \vec F &= m\cdot \left(\left(\ddot r - r \dot\theta^2\right)\hat r + \left(2 \dot r\dot \theta + r\ddot \theta\right)\hat \theta\right)\,. \end{align}

This is useful in any case where the resulting force is given in polar coordinates; in particular, a radial force $\vec F = F\hat r$ gives the equations \begin{align} F &= m\cdot \left(\ddot r- r\dot\theta^2\right)\\ 0 &= 2\dot r\dot \theta + r\ddot \theta\,. \end{align}

The second equation can be separated as \begin{align} -\frac{\ddot \theta}{\dot \theta} = 2\frac{\dot r}{r}\,, \end{align} which is integrated to give \begin{align} -\ln(\dot\theta) &= 2\ln(r) + \mathrm{const.} \end{align} or \begin{align} r^2\dot\theta &= \mathrm{const.} \end{align}

The quantity $r^2\dot\theta$ is conserved, but this is just the specific angular momentum magnitude: \begin{align} \vec L &= \vec r\times \vec p = m \vec r \times \vec v = m r^2 \dot\theta \hat z\,, \end{align} where $\hat z = \hat r\times\hat \theta$ is the constant vector perpendicular to the plane of movement. A particle moving in a radial force field thus moves with constant orbital angular momentum.

One consequence of this is Kepler's second law of planetary motion (the area law), as the area swept by the line segment between the orbiting particle and the center of its orbit in time $\mathrm dt$ is $\mathrm dA=r^2\dot\theta \,\mathrm dt$