Elastic reflection from a fixed sphere: the branch $b=a\cos (\theta \mathrm{/}2)$, the uniform density $a^2\mathrm{/}4$ and the total $\pi a^2$ of a contact model

Assume ${\mathrm{A}\mathrm{C}}_{\omega }$ (The Axiom of Countable Choice (${\mathrm{A}\mathrm{C}}_{\omega }$)) for the mathematical incident and solid-angle measures below.

Declared model. Fix a length $a>0$ (the sphere radius) and a reduced mass $\mu >0$. In a fixed inertial chart (Inertial frames, stationary observers and the adopted Galilean transformations between frames), let the sphere $S_a:=\{x\in {\mathbb{R}}^3:\parallel x{\parallel }_2=a\}$ be immovable and centred at the origin. For $b\in (0,a)$ the label position is a continuous curve ${\gamma }_b:\mathbb{R}\to {\mathbb{R}}^3$, affine on each side of one contact time $t_{\ast }$, with one-sided velocity limits and $P:={\gamma }_b(t_{\ast })\in S_a$. Its incoming affine line has speed $v_{\mathrm{\infty }}>0$, direction $\widehat{u}$ and perpendicular distance $b$ from the origin. Separately adopt, for this fixed-wall model, the elastic reflection rule at the contact point with outward unit normal $n:=P\mathrm{/}a$: $v^{+}=v^{-}-2⟨v^{-},n⟩n,$ that is, the normal component reverses and the tangential component is unchanged. Its normal-velocity reversal is analogous to the $e=1$ case of Adopted impact and open-mass extensions of the force branch: the isolated reciprocal impulse model, the collinear restitution contact law and the open-body material accounting clause (ii), but it is not a formal member of that law: the postulate assumes a finite-mass two-label collinear contact, whereas this example prescribes an immovable wall and a vector reflection rule. The inward unit normal $-n$ makes the angle $\alpha :=\arcsin \frac{b}{a}\in (0,\frac{\pi }{2})$ with $\widehat{u}$; equivalently $b=a\sin \alpha$ is the perpendicular distance of the incoming line from the centre.

Incident preparation and contact cross-section. Choose orthonormal vectors $e_1,e_2$ perpendicular to the common incoming direction $\widehat{u}$. For an incident offset $b{e}_{\phi }$, where $e_{\phi }=\cos \phi e_1+\sin \phi e_2$, the first contact is $P=b{e}_{\phi }-\sqrt{a^2-b^2}\widehat{u}$. The transverse component of the reflected direction is a positive multiple of $e_{\phi }$, so its outgoing azimuth is $\phi$. Let $D=(0,a)\times (0,2\pi )$ and $Y=(0,\pi )\times (0,2\pi )$. Adopt uniform incident area measure $\nu (B)={\int }_{B}bd{\lambda }_2(b,\phi )$ on Borel subsets of $D$ and solid-angle measure $\mathrm{\Omega }(B)={\int }_B\sin \theta d{\lambda }_2(\theta ,\phi )$ on $Y$, using the abstract mathematical measures of Scattering data: the incident flux measure, the solid-angle element, deflection branches and the cross-section density. Define the contact cross-section measure by $\sigma (B):=\nu (T^{-1}(B))$, where $T(b,\phi )=(\theta (b),\phi )$ is the outgoing direction map. For an angular band $A\subseteq (0,\pi )$ abbreviate $\sigma (A):=\sigma (A\times (0,2\pi ))$. A differential contact cross-section is a density $f$ satisfying $\sigma (B)={\int }_{B}fd\mathrm{\Omega }$ for every Borel $B\subseteq Y$. This is a local definition for the fixed-wall contact model, independent of the smooth central-model class of Scattering data of the central model class: the scattering motion, the impact parameter, the asymptotic directions, the deflection conventions and the uniform-incident preparation. The centre, grazing circle and azimuth seam have zero incident area and supply no atoms; free incoming lines with $b>a$ do not hit the sphere and are excluded from this contact cross-section.

Assertion to check. The motion is a single reflection with deflection $\theta (b)=\pi -2\alpha =\pi -2\arcsin \frac{b}{a}\in (0,\pi ),$ so the branch is $b(\theta )=a\cos (\theta \mathrm{/}2)$ on $0<\theta <\pi$, the differential cross-section is the constant $a^2\mathrm{/}4$, and the truncated total cross-section over the whole branch is $\pi a^2$.

• Units. $[a]=[b]=\mathrm{\ell }$, $[\mu ]=\mathsf{M}$, $[v_{\mathrm{\infty }}]=\mathrm{\ell }{\tau }^{-1}$, $[\mathrm{d}\sigma \mathrm{/}\mathrm{d}\mathrm{\Omega }]={\mathrm{\ell }}^2$, $[\sigma ]={\mathrm{\ell }}^2$, and $[j]=\mathsf{M}{\mathrm{\ell }}^2{\tau }^{-1}$ with $j=\mu b{v}_{\mathrm{\infty }}$ (Dimensions and SI unit conventions of the non-relativistic classical mechanics framework).

• Why the rule is not a potential. The impenetrable sphere corresponds to no $C^2$ finite-range potential; its force is a contact impulse at $\mathrm{\mid }x\mathrm{\mid }=a$. This example-specific fixed-wall law differs from the finite-mass two-label extension in Adopted impact and open-mass extensions of the force branch: the isolated reciprocal impulse model, the collinear restitution contact law and the open-body material accounting, so the example illustrates the model boundary rather than an instance of its restitution law or the smooth-potential theorem.