Section 21.10 Equations of Motion: \(F=\mu a\)
The problem is now to the point where we can write the equations of motion in a form we can solve. However, the importance of the preceding sections cannot be stressed enough. The strategies that we used are important to the success of problem solving in many complicated physics situations. Drawing a picture, exploiting symmetries, choosing a convenient origin, and using the most appropriate coordinate system all combine to make the analysis as easy as possible. These and other tricks should always be regarded as a good beginning to any problem.
Newton’s second law, reduced and modified for our specific problem is:
\begin{equation}
f(r)\,\rhat= \mu \ddot{\rr}
= \mu \left(
(\ddot{r} - r\dot\phi^2) \,\rhat
+ (r\ddot\phi + 2\dot{r}\dot\phi)\,\phat \right)\tag{21.10.1}
\end{equation}
The vector equation breaks up, in polar coordinates, into two coupled differential equations for \(r(t)\) and \(\phi(t)\text{:}\)
\begin{align}
f(r) \amp= \mu (\ddot{r} - r\dot\phi^2)\tag{21.10.2}\\
0 \amp= \mu (r\ddot\phi + 2\dot{r}\dot\phi)\tag{21.10.3}
\end{align}
(21.10.3) is just the polar coordinate statement of angular momentum conservation, which we have already discussed, i.e.:
\begin{equation}
0
= r \mu (r\ddot\phi + 2\dot{r}\dot\phi)
= \frac{d}{dt} \left(\mu r^2 \dot\phi \right)
= \frac{d\ell}{dt}\tag{21.10.4}
\end{equation}
(To derive verify the equalities in (21.10.4) it is easiest to work from right to left!) Therefore
\begin{equation}
\mu r^2 \dot\phi = \ell = \hbox{constant}\tag{21.10.5}
\end{equation}
can be solved for \(\dot\phi\) and used in (21.10.2) to obtain a messy, second-order ODE for \(r(t)\text{:}\)
\begin{equation}
\ddot{r} = \frac{\ell^2}{\mu^2r^3} + \frac{1}{\mu} f(r)\tag{21.10.6}
\end{equation}
In principle, we could now insert the particular form of \(f(r)\) we are concerned with, solve (21.10.6) for \(r\) as a function of \(t\text{,}\) and insert this value in (21.10.5) and solve for \(\phi(t)\text{.}\) We would then have solved the equations of motion for \(r\text{,}\) and \(\phi\text{,}\) parameterized by the time \(t\text{.}\) In practice, for any but the simplest forms of \(f(r)\text{,}\) it is impossible to solve the differential equations analytically. Computers to the rescue! In Section 21.16 you can explore numerical solutions for a \(1/r^2\) force.
