= Reflection symmetry of a two-sphere passing trajectory
{title2=$\mathbf r_j(t_c+\tau)=S\mathbf r_j(t_c-\tau)$}
In a uniform vertical tube, take two spherical particles driven by fixed axial <forces>, including a force-free particle as a special case. At a side-by-side passage time $t_c$, their centres lie in one horizontal reflection plane $S$. Reflection reverses the applied axial <forces>, and time reversal reverses them again. Uniqueness of the particle-velocity initial-value problem therefore identifies the reflected reverse trajectory with the original forward one. Their transverse positions before and after passage are equal at matching axial separations. The argument requires passage without contact and neglects inertia or other nonreversible effects.
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