= Non-spiralling overdamped central oscillator trajectory
{title2=$\Delta\phi_\infty=\arctan[\omega/(\gamma+\sqrt{\gamma^2-\omega^2})]<\pi/4$}
For the <initially circular damped planar oscillator> with <overdamping> $\gamma>\omega$, put $\alpha=\sqrt{\gamma^2-\omega^2}$. The exact solution replaces $\cos\nu t,\sin\nu t/\nu$ by $\cosh\alpha t,\sinh\alpha t/\alpha$. Resolving along the starting radius and its anticlockwise tangent yields
$$
\tan\Delta\phi(t)=\frac{\omega\sinh\alpha t}{\alpha\cosh\alpha t+\gamma\sinh\alpha t}.
$$
Its limiting angle is the title value. Both coordinates remain positive when $c>0$, so the trajectory returns to the origin without repeated revolutions. It initially moves tangentially and bends toward its slow decaying <eigenmode>, rather than following a radial straight line from the start.
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