Two-body reduction of a static central pair-potential model: the relative coordinate is a central-force trajectory for the reduced mass, the centre of mass moves uniformly, and the reconstruction is a bijection

Adopt the force model of Adopted force model: force maps, the equation of motion, additive composition of force contributions and frame covariance with $N=2$ labels, masses $m_1,m_2>0$ (Particle labels, worldlines, the frame-time parameter condition and fixed inertial masses clause (v)), a time window $I_0$, and the collision-excluded configuration domain $Q:=\{x=(x_1,x_2)\in ({\mathbb{R}}^3{)}^2:\mid x_1-x_2\mid \in J_0\}$ for an open interval $J_0\subseteq (0,\mathrm{\infty })$ (Configuration domain of $N$ labeled particles: open admissible sets, regular holonomic constraints and the mass operator). Suppose the model presents its force map through the time-independent potential $U(x_1,x_2):=u(\mid x_1-x_2\mid ),$ where $u:J_0\to \mathbb{R}$ is of class $C^2$ ($C^k$ Euclidean maps and diffeomorphisms), so that $F_i(t,x,v)=-{\mathrm{\nabla }}_{x_i}U(x)$ on the state domain (Restricted interaction models: pair decompositions, antisymmetry and centrality options, potential forces and the gravitational branch, clauses (iii) and (iv), The radial composite $x\mapsto u(\mid x\mid )$ on an annulus: $C^k$ regularity and $\mathrm{\nabla }(u\circ \mid \cdot \mid )(x)=u^{\mathrm{\prime }}(\mid x\mid )x\mathrm{/}\mid x\mid$). Let $x:J\to Q$ be a motion of the model on an interval $J$ with nonempty interior (The associated first-order system, motions, maximal motions and the Newtonian flow of a force model) and write $M:=m_1+m_2,\mu :=\frac{m_1{m}_2}{M},X:=\frac{m_1{x}_1+m_2{x}_2}{M},V:=X^{\mathrm{\prime }},P:=m_1{x}_1^{\mathrm{\prime }}+m_2{x}_2^{\mathrm{\prime }},r:=x_1-x_2,$ with $\mid \cdot \mid =\parallel \cdot {\parallel }_2$ the Euclidean norm (The Euclidean inner product $⟨x,y⟩={\sum }_{k<n}{x}_k{y}_k$ on ${\mathbb{R}}^n$) and dimensions as in Dimensions and SI unit conventions of the non-relativistic classical mechanics framework ($[M]=[\mu ]=\mu$, $[X]=[r]=\mathrm{\ell }$, $[P]=\mu \mathrm{\ell }{\tau }^{-1}$). Then:

1. (Pair forces.) On $J$ the evaluated forces are $F_1(t,x(t),x^{\mathrm{\prime }}(t))=-\frac{u^{\mathrm{\prime }}(\mid r\mid )}{\mid r\mid }r,F_2(t,x(t),x^{\mathrm{\prime }}(t))=+\frac{u^{\mathrm{\prime }}(\mid r\mid )}{\mid r\mid }r;$ thus the force map admits the external-plus-pair decomposition of Restricted interaction models: pair decompositions, antisymmetry and centrality options, potential forces and the gravitational branch clause (i) with $F_i^{\mathrm{e}\mathrm{x}\mathrm{t}}\equiv 0$ and the single pair force $F_{12}=-u^{\mathrm{\prime }}(\mid r\mid )r\mathrm{/}\mid r\mid =-F_{21}$, which is central and antisymmetric in the sense of clause (ii). This identity does not assert that a separate decomposition was declared as model data.
2. (Centre of mass.) $P$ is constant on $J$ and $X$ is affine: for all $t,t_0\in J$, $X(t)=X(t_0)+(t-t_0)V$ with the constant $V=X^{\mathrm{\prime }}=M^{-1}P$.
3. (Relative motion.) $r$ is of class $C^2$ on $J$, satisfies $\mu r^{\mathrm{\prime }\mathrm{\prime }}(t)=-\frac{u^{\mathrm{\prime }}(\mid r(t)\mid )}{\mid r(t)\mid }r(t)(t\in J),$ and is therefore a central-force trajectory for $(\mu ,u)$ in the sense of Central-force trajectories: the central-force equation, angular momentum, energy and the effective potential; its angular momentum vector $j=\mu r\times r^{\mathrm{\prime }}$ and energy $E=\frac{1}{2}\mu \parallel r^{\mathrm{\prime }}{\parallel }_2^2+u(\mid r\mid )$ are the attached objects of that definition.
4. (Reconstruction and bijection.) The identities $x_1=X+\frac{m_2}{M}r$ and $x_2=X-\frac{m_1}{M}r$ hold on $J$. Conversely, if $X^{\circ }:J\to {\mathbb{R}}^3$ is affine with $(X^{\circ }{)}^{\mathrm{\prime }\mathrm{\prime }}=0$ and $r^{\circ }:J\to {\mathbb{R}}^3$ is a $C^2$ curve with $\mid r^{\circ }(t)\mid \in J_0$ and $\mu (r^{\circ }{)}^{\mathrm{\prime }\mathrm{\prime }}=-u^{\mathrm{\prime }}(\mid r^{\circ }\mid )r^{\circ }\mathrm{/}\mid r^{\circ }\mid$ on $J$, then $x_1^{\circ }:=X^{\circ }+\frac{m_2}{M}{r}^{\circ },x_2^{\circ }:=X^{\circ }-\frac{m_1}{M}{r}^{\circ }$ define a motion $x^{\circ }:J\to Q$ of the model whose centre of mass is $X^{\circ }$ and whose relative coordinate is $r^{\circ }$. Consequently the assignment $x\mapsto (X,r)$ is a bijection from the motions of the model on $J$ onto the pairs (affine curve $X^{\circ }$ with $(X^{\circ }{)}^{\mathrm{\prime }\mathrm{\prime }}=0$!, central-force trajectory $r^{\circ }$ for $(\mu ,u)$ with values in $J_0$) on $J$.

• What is and is not assumed. The reduction uses the potential presentation of the model, hence the absence of external forces; the proof sums the two equations directly. If the model also declares the zero-external-force pair decomposition, Total momentum balance and centre-of-mass equation for a model with antisymmetric pair forces gives the same conclusion, but its declaration hypothesis is not inferred from the force identity. The companion counterexample shows that centrality of the pair forces alone, without the no-external-force hypothesis, does not yield the reduction.

• Scope of the reconstruction. The bijection is stated on a fixed interval $J$ with nonempty interior; it does not assert global existence of either side, and the collision set $x_1=x_2$ is excluded by the configuration domain, so $\mid r\mid >0$ throughout.

• No empirical content. The masses and the potential are model data; the result is a conditional deduction in the adopted force and interaction branches, and the reduced mass is the inertial combination $\mu =m_1{m}_2\mathrm{/}(m_1+m_2)$, distinct from any gravitational coupling parameter.