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\begin{document}
$\varpi$
\[\varpi\]
{\large $\varpi$}{\Large $\varpi$}{\LARGE $\varpi$}

$\infty\qquad\propto\quad\uparrow\quad\Uparrow\quad\downarrow\quad\Downarrow\quad\quad\equiv$
\[\propto\quad\uparrow\quad\Uparrow\quad\downarrow\quad\Downarrow\quad\]
\[ b=a\sqrt{1-e^2}\]
\[\tan\left({\phi\over2}\right) = \sqrt{1+e\over 1-e}\;\tan\left({u\over2}\right) \]
\[u-e\sin u=\omega t \qquad \mbox{where: } \omega=\sqrt{\alpha\over ma^3} \]
\[v_0=\sqrt{\alpha\over ma(1-e^2)}\]
\[ {\bf a}={d\phantom{t} \over dt}\,{\bf v}(t) = {1\over m}\,{\bf F}({\bf r}(t))=
-\;{\alpha\hat{\bf r}\over m r^2} \]
\[|{\bf a}|=v_0\dot{\phi}\]
\[L=mr^2\dot{\phi}=mv_0(1+e)\;a(1-e)=\sqrt{\alpha ma(1-e^2)}\]
\[L=\sqrt{\alpha ma(1-e^2)}\]
\begin{eqnarray*}
|{\bf a}|&=&v_0\dot{\phi}\\
&=&v_0\left({v_0a(1-e^2)\over r^2}\right)\\
&=&{\alpha\over mr^2}
\end{eqnarray*}
\[E=-\;{\alpha\over2 a}\]
\[V_{\mbox{eff}}(r)=U(r)+{L^2\over 2mr^2}\]
\[ r= {a(1-e^2)\over (1+ e \cos\phi)}\]
\[\mbox{impact parameter: } b=a\sqrt{e^2-1}\]
\[r={a(e^2-1)\over 1+e\cos\phi}\]
\[\tan\left({\phi\over2}\right) = \sqrt{e+1\over e-1}\;\tanh\left({u\over2}\right) \]
\[e\sinh u-u=\omega t \qquad \mbox{where: } \omega=\sqrt{\alpha\over ma^3} \]
\[v_0=\sqrt{\alpha\over ma(e^2-1)}={v_{\infty}\over\sqrt{e^2-1}}\]
\[L=mr^2\dot{\phi}=mv_{\infty}b=\sqrt{\alpha ma(e^2-1)}\]
\[{2E} = {\alpha\over a} = m v_{\infty}^2 \]
\[\cos\left(\phi_{\rm max}\right) = -\;{1\over e} = -\sin(\theta/2) \qquad \phi_{\rm max} = {\theta/2}+{\pi/2}\]
\[\tan\left({\theta\over2}\right)={v_0\over v_{\infty}} = {\alpha/b\over m v_{\infty}^2} \sim {\rm PE\over KE}\]
\[r=a(e\cosh(u)-1)\]
\[r=a(1-e\cos(u))\]

\end{document}
\bye
