Adopted central-force model class: static central pair potentials, the reduced one-label problem, the fixed-centre idealization, reading conventions and exclusions

Adopt, as the central-force model for the two-body part of this page, the following restricted model class inside the force and interaction branches of Adopted force model: force maps, the equation of motion, additive composition of force contributions and frame covariance and Restricted interaction models: pair decompositions, antisymmetry and centrality options, potential forces and the gravitational branch.

(i) The two-label central model. A model of class (i) is a force model of Adopted force model: force maps, the equation of motion, additive composition of force contributions and frame covariance with $N=2$ labels, fixed masses $m_1,m_2>0$ (Particle labels, worldlines, the frame-time parameter condition and fixed inertial masses clause (v)), a time window $I_0$, 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), and the force map presented through the time-independent static central pair potential $U(x_1,x_2):=u(\mid x_1-x_2\mid ),u:J_0\to \mathbb{R}\text{ of class }{C}^2,$ so that $F_i(t,x,v)=-{\mathrm{\nabla }}_{x_i}U(x)$ on the state domain (clauses (iii) and (iv) of Restricted interaction models: pair decompositions, antisymmetry and centrality options, potential forces and the gravitational branch; $U$ is of class $C^2$ on the open $Q$ by The radial composite $x\mapsto u(\mid x\mid )$ on an annulus: $C^k$ regularity and $\mathrm{\nabla }(u\circ \mid \cdot \mid )(x)=u^{\mathrm{\prime }}(\mid x\mid )x\mathrm{/}\mid x\mid$). In the external-plus-pair presentation of clause (i) of that postulate the model has $F_i^{\mathrm{e}\mathrm{x}\mathrm{t}}\equiv 0$: no external forces act on the labels of a model of class (i).

(ii) The reduced one-label problem. For a model of class (i) put $\mu :=m_1{m}_2\mathrm{/}(m_1+m_2)>0$ of dimension $\mu$ (Dimensions and SI unit conventions of the non-relativistic classical mechanics framework). The reduced problem of the model is the one-label problem of mass $\mu$ whose unknown is the relative coordinate $r=x_1-x_2$ and whose equation is $\mu r^{\mathrm{\prime }\mathrm{\prime }}=-\frac{u^{\mathrm{\prime }}(\mid r\mid )}{\mid r\mid }r;$ by 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 the relative coordinate of every motion of a model of class (i) solves this equation, and every pair of a uniform centre-of-mass motion and a solution of the reduced equation on a common interval reconstructs a motion of the model. Adopt the reduced problem as the object that the central-force statements of this page describe, and the central-force-trajectory vocabulary of Central-force trajectories: the central-force equation, angular momentum, energy and the effective potential (attached angular momentum $j=\mu r\times r^{\mathrm{\prime }}$ and energy $E=\frac{1}{2}\mu \parallel r^{\mathrm{\prime }}{\parallel }_2^2+u(\mid r\mid )$) as the vocabulary in which those statements are phrased.

(iii) The one-label fixed-centre idealization. A one-label central model is a force model of Adopted force model: force maps, the equation of motion, additive composition of force contributions and frame covariance with $N=1$, mass $\mu >0$, time window $I_0$, a fixed centre $c\in {\mathbb{R}}^3$, the configuration domain $Q_c=\{x\in {\mathbb{R}}^3:\mid x-c\mid \in J_0\}$, and the external force map $F(t,x,v)=-\frac{u^{\mathrm{\prime }}(\mid x-c\mid )}{\mid x-c\mid }(x-c),$ presented through the potential $u(\mid x-c\mid )$ with the same function $u\in C^2(J_0)$. This model is the fixed-centre idealization of the reduced problem: it drops the centre-of-mass degree of freedom and holds the force centre fixed. It is a separate adopted model class, not a consequence of class (i), and every use of it must state that the idealization is intended.

(iv) Reading conventions. In the statements of this page: the force centre of the reduced problem is the origin of the relative coordinate $r$; an orbit is the trajectory $t\mapsto r(t)$ of the reduced problem (equivalently, by 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, the trajectory of either label relative to the centre of mass, up to the mass ratio); "circular" and "elliptic" name properties of that trajectory; and the words "planet", "satellite", "central body" and "test particle" name labels of a model of class (i) or (iii). These words carry no empirical claim about nature.

(v) Exclusions and status. The class excludes: velocity-dependent and explicitly time-dependent central forces; non-central and non-potential forces; constraints and reaction forces; colliding configurations $x_1=x_2$ (equivalently $r=0$); external forces on the labels; and models with more than two labels. Finite-size, rotational, deformational, dissipative and relativistic effects are outside the class. The masses, the interval $J_0$ and the function $u$ (including any parameters it carries) are model data, not measured values (Dimensions and SI unit conventions of the non-relativistic classical mechanics framework); quantitative empirical use requires separately stated measured inputs with their uncertainties, and no statement of this page asserts that any physical system is exactly a model of this class.

• Why the class is adopted rather than derived. Centrality, the existence of a potential and the absence of external forces are restrictions of the force branch, and Restricted interaction models: pair decompositions, antisymmetry and centrality options, potential forces and the gravitational branch records that an arbitrary force map need satisfy none of them. This postulate selects the subclass the page develops, and every theorem of the page remains conditional on it.

• The reduction is a theorem, the class is not. The construction that turns a model of class (i) into the reduced problem is proved in 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; this postulate only names that object and fixes the vocabulary. The reader is not asked to accept the reduction as an assumption.

• The idealization is labelled. The fixed-centre model of clause (iii) is the model actually used when one body is treated as immovable; the exact two-label statement always retains both bodies through the relative coordinate and the centre of mass.