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\)
Section 21.15 Equations of Motion: \(E=T+U\)
Another theoretical tool we can use to arrive at an equation for the orbit is conservation of energy. The central force \(\FF\) is conservative and can be derived from a potential \(U(r)\) which depends only on the distance from the center of mass (see practice problem [1.2]):
\begin{equation}
\FF= -\grad U = -\frac{\partial U(r)}{\partial r}\,\rhat\tag{21.15.1}
\end{equation}
The statement of energy conservation:
\begin{equation}
E = T + U\tag{21.15.2}
\end{equation}
\begin{equation}
E = \frac12\mu \dot{r}^2 + \frac12\frac{\ell^2}{\mu r^2} + U(r)\tag{21.15.3}
\end{equation}
(21.15.3) can be solved for
\(\dot{r}\) to give:
\begin{equation}
\dot{r} =
\pm\sqrt{\frac{2}{\mu}\bigl(E-U(r)\bigr)-\frac{\ell^2}{\mu^2r^2}}\tag{21.15.4}
\end{equation}
(21.15.4) is an equivalent alternative to
(21.11.6) as an equation of motion for
\(r(t)\text{.}\) You might be surprised that
(21.15.4) is a first order differential equation, whereas
(21.11.6) is second order. This means that only one initial condition is required for the solution of
(21.15.4) whereas two are needed for the solution of
(21.11.6) . There is nothing surprising going on here. We have already provided the extra information (the extra initial condition) by specifying the constant total energy
\(E\text{.}\)
