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Two-body, central forces and gravitation

1 · Prerequisites

2 · Summary

This page develops the two-body problem with mutual central forces and its gravitational specialization in the framework of cm02-forces-initial-data-and-newtonian-flow and cm03-momentum-work-and-energy-balances, consuming the mathematical central-force development of cm-math-central. The local lemma The radial composite $x \mapsto u (∣ x ∣)$ on an annulus: $C^{k}$ regularity and $\nabla (u \circ ∣ \cdot ∣) (x) = u^{'} (∣ x ∣) x / ∣ x ∣$ supplies the gradient of a radial potential, $U (x) = u (∣ x ∣)$ with $\nabla U (x) = u^{'} (∣ x ∣) x / ∣ x ∣$ on an annulus, together with the $C^{k}$ regularity of the composite that the potential presentation requires. On that basis 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 proves, for a two-label model presented through a static central pair potential $u (∣ x_{1} - x_{2} ∣)$ with no external forces, that the evaluated pair forces are central and antisymmetric, that the total momentum and the centre-of-mass velocity are constant with an affine centre of mass, that the relative coordinate $r = x_{1} - x_{2}$ is a central-force trajectory for the reduced mass $μ = m_{1} m_{2} / (m_{1} + m_{2})$ , and that the motion is reconstructed bijectively from the centre-of-mass motion and the relative trajectory. The adopted model class, the reduced problem, the fixed-centre idealization and the reading conventions are fixed by the postulate Adopted central-force model class: static central pair potentials, the reduced one-label problem, the fixed-centre idealization, reading conventions and exclusions, and the companion counterexample on the examples page shows that centrality of the pair forces alone does not give the reduction without the no-external-force hypothesis.

The first consequences of the reduction are collected in Planarity of the reduced orbit and the area law: the areal-velocity vector is the constant $j / (2 μ)$ , the swept-area rate is $J / (2 μ)$ , and both labels sweep equal areas in equal times about the centre of mass: the planar character of the relative motion, the constant areal-velocity vector $j / (2 μ)$ , the linear swept-area function with rate $J / (2 μ)$ — Kepler's second law for the reduced orbit — and the constant areal rates of both labels about the centre of mass for $j \neq 0$ , in the ratio $(m_{2} / m_{1})^{2}$ . Reduction of the central problem to the effective one-dimensional potential: $μ ρ^{''} = - (U_{J}^{e f f})^{'} (ρ)$ , the energy identity, and the converse reconstruction of the planar trajectory from the radial data establishes the equivalent one-dimensional problem for nonzero angular momentum $J > 0$ : the radial coordinate takes values in $J_{0}$ and obeys $μ ρ^{''} = - (U_{J}^{e f f})^{'} (ρ)$ with the effective potential $U_{J}^{e f f} = u + J^{2} / (2 μ ρ^{2})$ , its energy is a constant first integral, and conversely every $C^{2}$ radial solution valued in $J_{0}$ together with the angular quadrature reconstructs a planar central-force trajectory, so the reduction to one dimension is lossless up to the angular phase. Circular relative orbits of the central model: the critical-point criterion $u^{'} (r_{0}) = J^{2} / (μ r_{0}^{3})$ , uniform rotation of both labels about the centre of mass, and the linearised radial frequency gives the critical-point criterion for a circular relative orbit in the equivalent form $μ r_{0} ω^{2} = u^{'} (r_{0})$ , the uniform rotation of both labels about the centre of mass, and the frequency of the linearised radial equation at a critical point where $(U_{J}^{e f f})^{''} (r_{0}) > 0$ , with no nonlinear stability claim.

The gravitational model is adopted in Adopted two-label gravitational model: the inverse-square pair potential with coupling parameters, the empirical mass identification, and the measured value and uncertainty of the Newtonian constant: the inverse-square pair potential with the Newtonian constant and positive gravitational coupling parameters, the empirical identification of the couplings with the inertial masses (the weak equivalence principle, adopted only where a statement says so), the measured value of the Newtonian constant with its uncertainty and source-reported input measurements, and the defined weak-field and slow-relative-motion regime. On that model The inverse-square gravitational relative trajectory has the Kepler polar form: $ρ = p / (1 + e \cos (θ - θ_{0}))$ , $E = (μ κ^{2} / 2 J^{2}) (e^{2} - 1)$ , elliptic completeness and barycentric scaled ellipses proves the Keplerian orbit classification: the polar conic with focus at the force centre, the eccentricity-energy relation and the elliptic, parabolic and hyperbolic branches, the centrifugal barrier that keeps every non-radial motion away from the collision, the completeness, closure and ellipse geometry of the negative-energy branch, and the scaled barycentric ellipses of both labels. Kepler's third law for the gravitational two-label model: the relative period is $T = 2 π μ a b / J = 2 π \sqrt{μ / κ} a^{3 / 2}$ , so $T^{2} = 4 π^{2} a^{3} / (G_{N} M)$ under the mass identification, and both labels are periodic relative to the centre of mass computes the period $T = 2 π μ a b / J = 2 π \sqrt{μ / κ} a^{3 / 2}$ , hence $T^{2} = 4 π^{2} a^{3} / (G_{N} M)$ under the mass identification, together with the barycentric periodicity of both labels and the area swept per period. The closing audit Scope of the two-body and central-force page: collision and radial exclusions, point-particle and finite-size limits, the empirical parameters and their uncertainty, and the conditional status of the orbit statements records the collision and radial exclusions, the point-particle and finite-size limits, the empirical status of $G_{N}$ and of the mass identification, and the claims deliberately not made.

Worked numerical checks of the elliptic branch, the parabolic and hyperbolic branches, the barycentric reconstruction with unequal masses, the circular orbit and its linearised frequency, and the counterexample showing the necessity of the no-external-force hypothesis are collected on cm05-two-body-central-forces-and-gravitation-examples.

3 · Logical flowchart

4 · Definitions, postulates, theorems and proofs

LemmaStatement: AI-adaptedProof: AI-adaptedOpen item page →

The radial composite $x \mapsto u (∣ x ∣)$ on an annulus: $C^{k}$ regularity and $\nabla (u \circ ∣ \cdot ∣) (x) = u^{'} (∣ x ∣) x / ∣ x ∣$

Statement

Let $J_{0} \subseteq (0, \infty)$ be an open interval (Intervals of $R$ : the nine order-convex forms, nondegeneracy, and length), let $u : J_{0} \to R$ be of class $C^{k}$ for some $k \geq 1$ ( $C^{k}$ Euclidean maps and diffeomorphisms), and put $A : = {x \in R^{3} : ∣ x ∣ \in J_{0}}, ∣ x ∣ : = ∥ x ∥_{2} = \sqrt{⟨ x, x ⟩},$ the Euclidean norm of The Euclidean inner product $⟨ x, y ⟩ = \sum_{k < n} x_{k} y_{k}$ on $R^{n}$ . Then:

$A$ is open in $R^{3}$ and $0 \notin A$ ;
the function $N : A \to (0, \infty)$ , $N (x) : = ∣ x ∣$ , is of class $C^{\infty}$ on $A$ ; it is totally differentiable at every $x \in A$ (The total (Fréchet) derivative $D f (a)$ as the linear first-order approximation with $o (∥ h ∥_{2})$ remainder) with $D N (x) h = \frac{⟨ x, h ⟩}{∣ x ∣} (h \in R^{3}), so \nabla N (x) = \frac{x}{∣ x ∣}$ (The Jacobian matrix of partial derivatives and the gradient in the scalar-valued case);
the radial composite $U : = u \circ N : A \to R$ , $U (x) = u (∣ x ∣)$ , is of class $C^{k}$ on $A$ and totally differentiable at every $x \in A$ with $D U (x) h = u^{'} (∣ x ∣) \frac{⟨ x, h ⟩}{∣ x ∣} = ⟨ u^{'} (∣ x ∣) \frac{x}{∣ x ∣}, h ⟩ (h \in R^{3}),$ that is, $\nabla U (x) = u^{'} (∣ x ∣) x / ∣ x ∣$ .

Facts & Assumptions

Given: The open interval $J_{0} \subseteq (0, \infty)$ , the $C^{k}$ function $u : J_{0} \to R$ with $k \geq 1$ , the domain $A = {x \in R^{3} : ∣ x ∣ \in J_{0}}$ , and the maps $N$ , $U$ of the statement.

[L1]

The Euclidean norm: $∣ x ∣^{2} = ⟨ x, x ⟩$ ; it vanishes only at $x = 0$ , is nonnegative everywhere, satisfies the triangle inequality $∣ x + y ∣ \leq ∣ x ∣ + ∣ y ∣$ , and $∣ λ x ∣ = ∣ λ ∣ ∣ x ∣$ (The Euclidean inner product $⟨ x, y ⟩ = \sum_{k < n} x_{k} y_{k}$ on $R^{n}$ , The induced length is a norm).

[L2]

Reverse triangle inequality: $∣ ∣ x ∣ - ∣ y ∣ ∣ \leq ∣ x - y ∣$ (The reverse triangle inequality in a normed space).

[L3]

Cauchy–Schwarz: $∣ ⟨ x, y ⟩ ∣ \leq ∣ x ∣ ∣ y ∣$ (Cauchy-Schwarz $∣ ⟨ x, y ⟩ ∣ \leq ∥ x ∥_{2} ∥ y ∥_{2}$ with its equality case, the triangle inequality for $∥ \cdot ∥_{2}$ , the parallelogram law and polarisation).

[L4]

Chain rule and total derivatives: a composite of totally differentiable maps is totally differentiable with the composed derivative (The chain rule for total derivatives: $D (g \circ f) (a) = D g (f (a)) \circ D f (a)$ ); the total derivative of a scalar-valued map at a point is the row $(D_{e_{j}} f (a))_{j}$ of directional derivatives, so on $R$ it is multiplication by the ordinary derivative (A total derivative computes every directional derivative, and its matrix is the Jacobian, Directional derivatives and partial derivatives of a map $U \subseteq R^{m} \to R^{n}$ , The Jacobian matrix of partial derivatives and the gradient in the scalar-valued case); a map whose first partial derivatives exist near a point and are continuous there is totally differentiable there (If all partial derivatives exist on a neighbourhood and are continuous at a point, then the map is totally differentiable there with Jacobian derivative); and for a totally differentiable scalar field $D_{v} f (a) = ⟨ \nabla f (a), v ⟩$ (For a differentiable scalar field, $D_{v} f (a) = ⟨ \nabla f (a), v ⟩$ and the unit direction of steepest ascent is the normalized gradient).

[L5]

$C^{k}$ closure: finite componentwise sums, products and scalar multiples of $C^{k}$ Euclidean maps are $C^{k}$ , and a composite of composable $C^{k}$ Euclidean maps is $C^{k}$ ; a map is $C^{\infty}$ when it is $C^{k}$ for every $k$ ( $C^{k}$ Euclidean maps are closed under componentwise algebra and composition, $C^{k}$ Euclidean maps and diffeomorphisms, $C^{k}$ maps and multi-index derivative notation in Euclidean space).

[L6]

Real powers: $x \mapsto x^{α}$ is continuous and differentiable on $(0, \infty)$ with derivative $α x^{α - 1}$ (Continuity and derivatives of positive-base real powers, Real powers for positive bases, with the zero-base positive-exponent convention).

[L7]

Induction principle for the statement "for every natural $m$ " (The principle of mathematical induction).

Proof

technique · direct

1.1givenL1L2algebra

The norm is continuous on $R^{3}$ by the reverse triangle inequality [L2], so $A$ , the preimage of the open set $J_{0}$ under $N$ , is open; and $N (0) = 0 \notin J_{0}$ gives $0 \notin A$ . This is clause 1.

1.2givenL1algebra

Fix $x \in A$ and $h \in R^{3}$ with $x + h \in A$ . By bilinearity, symmetry and [L1], $∣ x + h ∣^{2} - ∣ x ∣^{2} = 2 ⟨ x, h ⟩ + ∣ h ∣^{2}$ , and therefore $∣ x + h ∣ - ∣ x ∣ = (2 ⟨ x, h ⟩ + ∣ h ∣^{2}) / (∣ x + h ∣ + ∣ x ∣)$ , the denominator being at least $∣ x ∣ > 0$ .

2.1step 1.2L1L2L3algebra

Remainder estimate at $x$ : subtracting $⟨ x, h ⟩ / ∣ x ∣$ from step 1.2 and putting over the common denominator $∣ x ∣ (∣ x + h ∣ + ∣ x ∣)$ gives $∣ ∣ x + h ∣ - ∣ x ∣ - \frac{⟨ x, h ⟩}{∣ x ∣} ∣ \leq \frac{∣ x ∣ ∣ h ∣^{2} + ∣ ⟨ x, h ⟩ ∣ ∣ ∣ x ∣ - ∣ x + h ∣ ∣}{∣ x ∣ (∣ x + h ∣ + ∣ x ∣)} \leq \frac{2 ∣ h ∣^{2}}{∣ x ∣},$ where [L3] bounds the first numerator term by $∣ x ∣ ∣ h ∣^{2}$ , [L2] bounds the second by $∣ x ∣ ∣ h ∣ \cdot ∣ h ∣$ , and the denominator is at least $∣ x ∣^{2}$ . Hence the normalized remainder tends to $0$ as $h \to 0$ : $N$ is totally differentiable at $x$ with $D N (x) h = ⟨ x, h ⟩ / ∣ x ∣$ .

3.1step 2.1L4L5algebra

The representing vector: for every $v$ , [L4] gives $D_{v} N (x) = D N (x) v = ⟨ x, v ⟩ / ∣ x ∣ = ⟨ x / ∣ x ∣, v ⟩$ ; comparing with $D_{v} N (x) = ⟨ \nabla N (x), v ⟩$ and taking $v$ over the standard basis shows $\nabla N (x) = x / ∣ x ∣$ . In particular every first partial derivative of $N$ agrees on $A$ with a component of $x / ∣ x ∣$ and is continuous, so $N$ is $C^{1}$ by $C^{k}$ Euclidean maps and diffeomorphisms.

3.2step 2.1L4algebra

Chain rule for the composite $U = u \circ N$ : $u$ is $C^{k}$ hence totally differentiable, $N$ is totally differentiable at every point of $A$ by step 2.1, and the domains compose, so [L4] gives $D U (x) = D u (N (x)) \circ D N (x)$ ; and $D U (x) h = u^{'} (∣ x ∣) \cdot D N (x) h = u^{'} (∣ x ∣) ⟨ x, h ⟩ / ∣ x ∣$ because the total derivative of the real function $u$ at $t$ is multiplication by $u^{'} (t)$ by [L4]. This is the displayed formula for $D U$ .

4.1step 3.1L5L6L7algebra

The induction on smoothness: the base case $m = 1$ is step 3.1. Assume $m \geq 1$ and $N \in C^{m} (A)$ . The map $t \mapsto t^{- 1}$ is $C^{\infty}$ on $(0, \infty)$ by [L6] with $α = - 1$ iterated, so $x \mapsto 1 / ∣ x ∣$ is $C^{m}$ on $A$ as the composite $t \mapsto t^{- 1}$ after $N$ . The coordinate maps $x \mapsto x_{j}$ are the components of the identity, whose partial derivatives are the constants $δ_{j k}$ ; they are therefore $C^{\infty}$ , hence $C^{m}$ , by $C^{k}$ Euclidean maps and diffeomorphisms and Directional derivatives and partial derivatives of a map $U \subseteq R^{m} \to R^{n}$ . So $\nabla N (x) = x / ∣ x ∣$ , a finite product of $C^{m}$ maps, is $C^{m}$ ; since a map is $C^{m + 1}$ exactly when its first partial derivatives are $C^{m}$ ( $C^{k}$ maps and multi-index derivative notation in Euclidean space), $N$ is $C^{m + 1}$ . By induction [L7], $N \in C^{m} (A)$ for every $m$ , that is $N \in C^{\infty} (A)$ . This completes clause 2.

5.1step 3.1step 3.2L4L5algebra∎

The gradient and the regularity: the display of step 3.2 writes $D U (x)$ as the linear map $h \mapsto ⟨ u^{'} (∣ x ∣) x / ∣ x ∣, h ⟩$ , so $\nabla U (x) = u^{'} (∣ x ∣) x / ∣ x ∣$ by the same uniqueness argument as step 3.1; and $U$ , a composite of the $C^{k}$ maps $u$ and $N$ ( $C^{k}$ Euclidean maps and diffeomorphisms), is $C^{k}$ on $A$ by [L5]. This is clause 3 and completes the proof.

Remarks

Why the annulus. The proof uses $∣ x ∣ \geq c > 0$ only through the open interval $J_{0} \subseteq (0, \infty)$ ; on a domain meeting the origin the norm is not differentiable there and the radial composite need not be. The collision-excluded configuration domains of the two-body model are exactly annuli of this shape.
The inverse-square instance. For $u (ρ) = - κ / ρ$ with $κ > 0$ on $J_{0} = (0, \infty)$ , clause 3 gives $\nabla U (x) = κ x / ∣ x ∣^{3}$ , so the potential $U (x) = - κ / ∣ x ∣$ presents the attractive inverse-square force $- k x / ∣ x ∣^{3}$ with $k = κ$ , the form used by the gravitational model of this batch.

Physics theoremStatement: AI-adaptedProof: AI-adaptedOpen item page →

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

Physical scope

The conditional reduction of a two-label model whose force map is presented by a static central pair potential: along every motion the evaluated pair forces are central and antisymmetric, the total momentum and the centre-of-mass velocity are constant, the relative coordinate satisfies the one-body central-force equation with the reduced mass, and the motion is reconstructed from the centre-of-mass motion and the relative trajectory. The result is a deduction inside the adopted force and interaction branches; it uses no measured value, assumes no external forces, excludes collisions, and asserts no conservation law beyond the vanishing of the total external force that the presentation supplies.

Statement

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 $Q : = {x = (x_{1}, x_{2}) \in (R^{3})^{2} : ∣ x_{1} - x_{2} ∣ \in J_{0}}$ for an open interval $J_{0} \subseteq (0, \infty)$ (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 $U (x_{1}, x_{2}) : = u (∣ x_{1} - x_{2} ∣),$ where $u : J_{0} \to R$ is of class $C^{2}$ ( $C^{k}$ Euclidean maps and diffeomorphisms), so that $F_{i} (t, x, v) = - \nabla_{x_{i}} U (x)$ 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 $x \mapsto u (∣ x ∣)$ on an annulus: $C^{k}$ regularity and $\nabla (u \circ ∣ \cdot ∣) (x) = u^{'} (∣ x ∣) x / ∣ x ∣$ ). 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 $M : = m_{1} + m_{2}, μ : = \frac{m_{1} m_{2}}{M}, X : = \frac{m_{1} x_{1} + m_{2} x_{2}}{M}, V : = X^{'}, P : = m_{1} x_{1}^{'} + m_{2} x_{2}^{'}, r : = x_{1} - x_{2},$ with $∣ \cdot ∣ = ∥ \cdot ∥_{2}$ the Euclidean norm (The Euclidean inner product $⟨ x, y ⟩ = \sum_{k < n} x_{k} y_{k}$ on $R^{n}$ ) and dimensions as in Dimensions and SI unit conventions of the non-relativistic classical mechanics framework ( $[M] = [μ] = μ$ , $[X] = [r] = ℓ$ , $[P] = μ ℓ τ^{- 1}$ ). Then:

(Pair forces.) On $J$ the evaluated forces are $F_{1} (t, x (t), x^{'} (t)) = - \frac{u^{'} (∣ r ∣)}{∣ r ∣} r, F_{2} (t, x (t), x^{'} (t)) = + \frac{u^{'} (∣ r ∣)}{∣ r ∣} r;$ 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 $F_{i}^{e x t} \equiv 0$ and the single pair force $F_{12} = - u^{'} (∣ r ∣) r / ∣ r ∣ = - F_{21}$ , 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.
(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 $V = X^{'} = M^{- 1} P$ .
(Relative motion.) $r$ is of class $C^{2}$ on $J$ , satisfies $μ r^{''} (t) = - \frac{u^{'} (∣ r (t) ∣)}{∣ r (t) ∣} r (t) (t \in J),$ and is therefore a central-force trajectory for $(μ, u)$ in the sense of Central-force trajectories: the central-force equation, angular momentum, energy and the effective potential; its angular momentum vector $j = μ r \times r^{'}$ and energy $E = \frac{1}{2} μ ∥ r^{'} ∥_{2}^{2} + u (∣ r ∣)$ are the attached objects of that definition.
(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 $X^{\circ} : J \to R^{3}$ is affine with $(X^{\circ})^{''} = 0$ and $r^{\circ} : J \to R^{3}$ is a $C^{2}$ curve with $∣ r^{\circ} (t) ∣ \in J_{0}$ and $μ (r^{\circ})^{''} = - u^{'} (∣ r^{\circ} ∣) r^{\circ} / ∣ r^{\circ} ∣$ 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 $(X^{\circ})^{''} = 0$ !, central-force trajectory $r^{\circ}$ for $(μ, u)$ with values in $J_{0}$ ) on $J$ .

Facts & Assumptions

Given: The two-label model with the $C^{2}$ static central pair potential $u$ on $J_{0}$ , the notation $M, μ, X, V, P, r$ of the statement, and the motion $x : J \to Q$ with coordinate velocities $w_{i} = x_{i}^{'}$ .

[L1]

Model notions (Adopted force model: force maps, the equation of motion, additive composition of force contributions and frame covariance clauses (i)-(iv), Restricted interaction models: pair decompositions, antisymmetry and centrality options, potential forces and the gravitational branch clauses (i)-(iv), The associated first-order system, motions, maximal motions and the Newtonian flow of a force model, Total mass, total momentum, centre of mass, angular momentum, torque, kinetic energy, power and work of a particle model in an inertial frame): the motion satisfies $m_{i} x_{i}^{''} (t) = F_{i} (t, x (t), w (t))$ for $t \in J$ ; the potential presentation is the identity $F_{i} (t, x, v) = - \nabla_{x_{i}} U (x)$ with $U$ of class $C^{2}$ on the open $Q$ ; and the quantities $M$ , $μ$ , $X$ , $V$ , $P$ , $r$ are those of the displayed definitions.

[L2]

Radial gradient (The radial composite $x \mapsto u (∣ x ∣)$ on an annulus: $C^{k}$ regularity and $\nabla (u \circ ∣ \cdot ∣) (x) = u^{'} (∣ x ∣) x / ∣ x ∣$ , clause 3): for the $C^{2}$ function $u$ on the open interval $J_{0} \subseteq (0, \infty)$ and $G (y) : = u (∣ y ∣)$ on $A = {y : ∣ y ∣ \in J_{0}}$ , the map $G$ is $C^{2}$ with $D G (y) ℓ = u^{'} (∣ y ∣) ⟨ y / ∣ y ∣, ℓ ⟩$ , that is $\nabla G (y) = u^{'} (∣ y ∣) y / ∣ y ∣$ .

[L3]

Chain rule (The chain rule for total derivatives: $D (g \circ f) (a) = D g (f (a)) \circ D f (a)$ ): composable totally differentiable maps compose with $D (G \circ S) (x) = D G (S (x)) \circ D S (x)$ ; the affine map $S (x_{1}, x_{2}) : = x_{1} - x_{2}$ has $D S (x) (h_{1}, h_{2}) = h_{1} - h_{2}$ , since $S (x + h) - S (x) = h_{1} - h_{2}$ is exactly linear in $h$ . The Euclidean inner product is bilinear (The Euclidean inner product $⟨ x, y ⟩ = \sum_{k < n} x_{k} y_{k}$ on $R^{n}$ ).

[L5]

Curve calculus (Differentiation of sums, scalar multiples, inner products, cross products and squared norms of differentiable curves in $R^{3}$ clauses 1 and 4, The derivative and the Riemann integral of a vector-valued function: an intrinsic derivative and a componentwise integral): sums and scalar multiples of differentiable curves are differentiable with the expected derivatives, so differences and second derivatives of $C^{2}$ curves are computed componentwise; and $t \mapsto ∥ y (t) ∥_{2}^{2}$ differentiates by the product rule.

[L6]

Vector FTC (If $f : [a, b] \to R^{m}$ is differentiable with integrable $f^{'}$ then $\int_{a}^{b} f^{'} = f (b) - f (a)$ ; and a bounded derivative makes $f$ Lipschitz): if $f : [a, b] \to R^{3}$ is differentiable with integrable derivative, then $\int_{a}^{b} f^{'} = f (b) - f (a)$ ; the interval $J$ has nonempty interior, hence at least two points, and the open annulus $A \subseteq R^{3}$ and the configuration domain $Q$ are those of the hypotheses (Intervals of $R$ : the nine order-convex forms, nondegeneracy, and length, Configuration domain of $N$ labeled particles: open admissible sets, regular holonomic constraints and the mass operator).

Proof

technique · direct

1.1givenL1L5

Setup: $x$ is a $C^{2}$ curve family with values in the open $Q$ , $∣ r (t) ∣ \in J_{0}$ for every $t$ , $r$ is $C^{2}$ by [L5], and $M > 0$ , $μ > 0$ , and the curves $X, V, P$ are defined; the evaluated gradient $\nabla_{x_{i}} U (x (t))$ is defined because $U$ is $C^{1}$ on the open $Q$ and $x (t) \in Q$ .

2.1step 1.1L1L2L3algebra

Pair forces: put $S (x_{1}, x_{2}) : = x_{1} - x_{2}$ and $G (y) : = u (∣ y ∣)$ on $A$ , so that $U = G \circ S$ . By [L2] and [L3], $D U (x) (h_{1}, h_{2}) = D G (r) (h_{1} - h_{2}) = u^{'} (∣ r ∣) ⟨ r / ∣ r ∣, h_{1} - h_{2} ⟩ = ⟨ u^{'} (∣ r ∣) r / ∣ r ∣, h_{1} ⟩ - ⟨ u^{'} (∣ r ∣) r / ∣ r ∣, h_{2} ⟩$ ; reading off the partial gradients gives $\nabla_{x_{1}} U (x) = u^{'} (∣ r ∣) r / ∣ r ∣$ and $\nabla_{x_{2}} U (x) = - u^{'} (∣ r ∣) r / ∣ r ∣$ . The equation of motion [L1] therefore gives $F_{1} (t, x (t), w (t)) = m_{1} x_{1}^{''} (t) = - u^{'} (∣ r ∣) r / ∣ r ∣$ and $F_{2} (t, x (t), w (t)) = + u^{'} (∣ r ∣) r / ∣ r ∣$ , and the presentation with $F_{1}^{e x t} = F_{2}^{e x t} = 0$ and $F_{12} : = - u^{'} (∣ r ∣) r / ∣ r ∣$ , $F_{21} : = - F_{12}$ is an identity for the force map; $F_{21} = - F_{12}$ is antisymmetry and both pair forces are parallel to $r = x_{1} - x_{2}$ , which is centrality. This is clause 1.

3.1step 2.1L1L5L6algebra

Centre of mass: by [L5] and step 2.1, $P^{'} = m_{1} x_{1}^{''} + m_{2} x_{2}^{''} = F_{1} + F_{2} = 0$ on $J$ . For $t_{0} < t$ in $J$ , [L6] applied to the $C^{1}$ curve $P$ on $[t_{0}, t]$ gives $P (t) - P (t_{0}) = \int_{t_{0}}^{t} P^{'} = 0$ ; reversing endpoints covers $t < t_{0}$ , so $P$ is constant. Differentiating the definition of $X$ using [L5] gives $X^{'} = M^{- 1} P = : V$ , hence $V$ is constant and $M X^{''} = 0$ . Applying [L6] to $X$ gives $X (t) - X (t_{0}) = \int_{t_{0}}^{t} V = (t - t_{0}) V$ for $t_{0} < t$ , and the reversed formula for $t < t_{0}$ . This proves clause 2 directly from the potential force map, without requiring a separately declared external-plus-pair presentation.

3.2step 1.1step 2.1L1L5algebra

Relative equation: by [L5], $r = x_{1} - x_{2}$ is $C^{2}$ with $r^{''} = x_{1}^{''} - x_{2}^{''}$ , so step 2.1 and $M = m_{1} + m_{2}$ give $μ r^{''} = - μ (\frac{1}{m_{1}} + \frac{1}{m_{2}}) \frac{u^{'} (∣ r ∣)}{∣ r ∣} r = - \frac{u^{'} (∣ r ∣)}{∣ r ∣} r$ , because $μ (\frac{1}{m_{1}} + \frac{1}{m_{2}}) = \frac{m_{1} m_{2}}{M} \cdot \frac{M}{m_{1} m_{2}} = 1$ . Since $r$ is $C^{2}$ on the interval $J$ with nonempty interior and $∣ r (t) ∣ \in J_{0}$ for every $t \in J$ , all hypotheses of Central-force trajectories: the central-force equation, angular momentum, energy and the effective potential hold with $(μ, U)$ replaced by $(μ, u)$ and the equation just displayed: $r$ is a central-force trajectory for $(μ, u)$ , and $j = μ r \times r^{'}$ , $E = \frac{1}{2} μ ∥ r^{'} ∥_{2}^{2} + u (∣ r ∣)$ are its attached objects. This is clause 3.

4.1step 1.1step 2.1step 3.2L1L5algebra∎

Reconstruction: the linear system $X = M^{- 1} (m_{1} x_{1} + m_{2} x_{2})$ , $r = x_{1} - x_{2}$ has the unique solution $x_{1} = X + \frac{m_{2}}{M} r$ , $x_{2} = X - \frac{m_{1}}{M} r$ on $J$ , which is the first assertion. Conversely, let $X^{\circ}$ be affine on $J$ with $(X^{\circ})^{''} = 0$ and let $r^{\circ}$ be a $C^{2}$ curve with $∣ r^{\circ} ∣ \in J_{0}$ and $μ (r^{\circ})^{''} = - u^{'} (∣ r^{\circ} ∣) r^{\circ} / ∣ r^{\circ} ∣$ . Define $x_{1}^{\circ}, x_{2}^{\circ}$ by the displayed formulas; then $x^{\circ}$ has values in $Q$ because $∣ x_{1}^{\circ} - x_{2}^{\circ} ∣ = ∣ r^{\circ} ∣ \in J_{0}$ , is $C^{2}$ , and $M^{- 1} (m_{1} x_{1}^{\circ} + m_{2} x_{2}^{\circ}) = X^{\circ}$ , $x_{1}^{\circ} - x_{2}^{\circ} = r^{\circ}$ . Moreover $m_{1} (x_{1}^{\circ})^{''} = m_{1} \frac{m_{2}}{M} (r^{\circ})^{''} = - \frac{u^{'} (∣ r^{\circ} ∣)}{∣ r^{\circ} ∣} r^{\circ} = F_{1}$ and $m_{2} (x_{2}^{\circ})^{''} = - m_{2} \frac{m_{1}}{M} (r^{\circ})^{''} = + \frac{u^{'} (∣ r^{\circ} ∣)}{∣ r^{\circ} ∣} r^{\circ} = F_{2}$ by step 2.1, so $x^{\circ}$ is a motion with centre of mass $X^{\circ}$ and relative coordinate $r^{\circ}$ . The two constructions are mutually inverse by the uniqueness in the first assertion, so $x \mapsto (X, r)$ is a bijection onto the stated pairs. This is clause 4 and completes the proof.

Remarks

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 $∣ r ∣ > 0$ 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 $μ = m_{1} m_{2} / (m_{1} + m_{2})$ , distinct from any gravitational coupling parameter.

PostulateStatement: AI-adaptedProof: Not applicableOpen item page →

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

Physical scope

Adoption of the page's central-force model class inside the framework's force and interaction branches: two-label models presented by a static $C^{2}$ central pair potential on a collision-excluded annulus and free of external forces; the reduced one-label problem of mass $μ$ ; the separate one-label fixed-centre idealization; the reading conventions for force centre, orbit and body labels; and the explicit exclusions (velocity- or time-dependent, non-central, non-potential, constrained, colliding, external-force and many-body models) together with the status of the model data. The item adopts a modelling class; it asserts no law of nature, no measured value and no conservation law.

Postulate

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 $Q : = {x = (x_{1}, x_{2}) \in (R^{3})^{2} : ∣ x_{1} - x_{2} ∣ \in J_{0}}$ for an open interval $J_{0} \subseteq (0, \infty)$ (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 $U (x_{1}, x_{2}) : = u (∣ x_{1} - x_{2} ∣), u : J_{0} \to R of class C^{2},$ so that $F_{i} (t, x, v) = - \nabla_{x_{i}} U (x)$ 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 $x \mapsto u (∣ x ∣)$ on an annulus: $C^{k}$ regularity and $\nabla (u \circ ∣ \cdot ∣) (x) = u^{'} (∣ x ∣) x / ∣ x ∣$ ). In the external-plus-pair presentation of clause (i) of that postulate the model has $F_{i}^{e x t} \equiv 0$ : 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 $μ : = m_{1} m_{2} / (m_{1} + m_{2}) > 0$ of dimension $μ$ (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 $μ$ whose unknown is the relative coordinate $r = x_{1} - x_{2}$ and whose equation is $μ r^{''} = - \frac{u^{'} (∣ r ∣)}{∣ r ∣} r;$ 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 $j = μ r \times r^{'}$ and energy $E = \frac{1}{2} μ ∥ r^{'} ∥_{2}^{2} + u (∣ r ∣)$ ) 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 $μ > 0$ , time window $I_{0}$ , a fixed centre $c \in R^{3}$ , the configuration domain $Q_{c} = {x \in R^{3} : ∣ x - c ∣ \in J_{0}}$ , and the external force map $F (t, x, v) = - \frac{u^{'} (∣ x - c ∣)}{∣ x - c ∣} (x - c),$ presented through the potential $u (∣ x - c ∣)$ 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.

Status and scope

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.

Physics theoremStatement: AI-adaptedProof: AI-adaptedOpen item page →

Planarity of the reduced orbit and the area law: the areal-velocity vector is the constant $j / (2 μ)$ , the swept-area rate is $J / (2 μ)$ , and both labels sweep equal areas in equal times about the centre of mass

Physical scope

The planar character of the reduced two-body orbit and the areal law, conditional on the adopted central-force model class: the attached angular-momentum vector is constant and orthogonal to the trajectory; for non-zero angular momentum the areal-velocity vector equals the constant $j / (2 μ)$ and the swept-area function grows linearly; and each label sweeps a constant areal rate about the centre of mass with the mass-ratio factor $(m_{2} / m_{1})^{2}$ . The result is a conditional deduction from the adopted model; it reports no observation, no measured value and no conservation beyond the conserved angular momentum of the central model.

Statement

(Plane.) $j$ is constant on $I$ , and $⟨ r (t), j ⟩ = 0$ and $⟨ r^{'} (t), j ⟩ = 0$ for every $t \in I$ ; if $j \neq 0$ , the relative trajectory is contained in the fixed plane ${y \in R^{3} : ⟨ y, j ⟩ = 0}$ through the force centre with normal $j$ .
(Areal velocity.) Suppose $j \neq 0$ and write $J : = ∣ j ∣ > 0$ . Let $(ρ, θ)$ be polar data for $r$ in an oriented orthonormal triad with $e_{3} = j / J$ (Polar reduction of a non-radial central-force trajectory: $ρ^{2} θ^{'} = J / μ$ , the effective-potential equation and energy conservation). Then $ρ^{2} θ^{'} = J / μ > 0$ , and the areal-velocity vector $ω (t) : = \frac{1}{2} r (t) \times r^{'} (t)$ is constant and equal to $j / (2 μ)$ ; in particular $∣ ω (t) ∣ = J / (2 μ)$ for all $t$ .
(Equal areas in equal times.) With $(ρ, θ)$ as in clause 2 and $t_{0} \in I$ , the swept-area function $A (t) : = \frac{1}{2} \int_{t_{0}}^{t} ρ (s)^{2} θ^{'} (s) d s (t \in I)$ is of class $C^{1}$ with $A^{'} \equiv J / (2 μ)$ ; hence for all $t_{0} < t_{1}$ in $I$ one has $A (t_{1}) - A (t_{0}) = (t_{1} - t_{0}) J / (2 μ)$ .
(Both labels about the centre of mass.) With $j \neq 0$ as in clause 2, for every $t \in I$ , $\frac{1}{2} (x_{1} (t) - X (t)) \times (x_{1} (t) - X (t))^{'} = (\frac{m_{2}}{M})^{2} ω (t), \frac{1}{2} (x_{2} (t) - X (t)) \times (x_{2} (t) - X (t))^{'} = (\frac{m_{1}}{M})^{2} ω (t),$ so each label's areal-velocity vector about the centre of mass is constant; their magnitudes are in the ratio $(m_{2} / m_{1})^{2}$ .

Facts & Assumptions

Given: The central-force model of class (i) with masses $m_{1}, m_{2}$ , the motion $x : I \to Q$ on the time interval $I$ , its centre of mass $X$ , relative coordinate $r$ , and attached angular momentum $j = μ r \times r^{'}$ ; where clause 2 or 3 is used, $j \neq 0$ and $(ρ, θ)$ is polar data in an oriented orthonormal triad with $e_{3} = j / J$ , $J = ∣ j ∣$ .

[L1]

Model and reduction (Adopted central-force model class: static central pair potentials, the reduced one-label problem, the fixed-centre idealization, reading conventions and exclusions clauses (i)-(ii), 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 clauses 1-4, Central-force trajectories: the central-force equation, angular momentum, energy and the effective potential): the relative coordinate is a central-force trajectory for $(μ, u)$ with attached angular momentum $j = μ r \times r^{'}$ and energy $E$ ; $X$ is affine with constant $V = X^{'}$ , and $x_{1} = X + \frac{m_{2}}{M} r$ , $x_{2} = X - \frac{m_{1}}{M} r$ on $I$ .

[L2]

Angular momentum (Conservation of the angular momentum vector along a central-force trajectory and its orthogonality to position and velocity): along a central-force trajectory, $j$ is differentiable with $j^{'} (t) = 0$ on $I$ , hence constant, and $⟨ r (t), j (t) ⟩ = ⟨ r^{'} (t), j (t) ⟩ = 0$ for all $t$ .

[L3]

Polar reduction (Polar reduction of a non-radial central-force trajectory: $ρ^{2} θ^{'} = J / μ$ , the effective-potential equation and energy conservation, Oriented orthonormal triads in $R^{3}$ : existence over any unit vector, and the orthonormal expansion with no fourth orthogonal direction): if $j = J e_{3}$ with $J > 0$ , then there are twice differentiable $ρ : I \to (0, \infty)$ and $θ : I \to R$ with $r = ρ (\cos θ e_{1} + \sin θ e_{2})$ and $ρ^{2} θ^{'} = J / μ$ ; the polar coordinates are those of the cited item.

[L4]

Cross-product and derivative rules (The cross product in $R^{3}$ , The cross product is bilinear, alternating, and orthogonal to both factors, Differentiation of sums, scalar multiples, inner products, cross products and squared norms of differentiable curves in $R^{3}$ , The derivative and the Riemann integral of a vector-valued function: an intrinsic derivative and a componentwise integral): the cross product is bilinear with $u \times u = 0$ ; sums, scalar multiples and differences of differentiable curves differentiate componentwise, so $j = μ r \times r^{'}$ and $(x_{i} - X)^{'} = x_{i}^{'} - X^{'}$ are computed by bilinearity and the product rule.

[L5]

Calculus of the integral (Every continuous function on an interval has a primitive; two primitives differ by a constant; and $\int_{a}^{b} f = G (b) - G (a)$ for any primitive $G$ , Newton–Leibniz needs only continuity on $[a, b]$ , differentiability on $(a, b)$ , and a Riemann-integrable extension of the interior derivative, Sums, scalar multiples, products, absolute values, maxima, minima and quotients with nonvanishing denominator of continuous functions are continuous, as are constants, the identity and every polynomial function, Intervals of $R$ : the nine order-convex forms, nondegeneracy, and length): a continuous function on an interval has a primitive, differentiable with that derivative; and if $G$ is continuous on $[a, b]$ , differentiable on $(a, b)$ with integrable derivative, then $\int_{a}^{b} G^{'} = G (b) - G (a)$ .

Proof

technique · direct

1.1givenL1L2algebra

Clause 1: by [L1] the curve $r$ is a central-force trajectory for $(μ, u)$ , so [L2] gives $j^{'} = 0$ on $I$ , hence $j$ constant, and $⟨ r (t), j ⟩ = ⟨ r^{'} (t), j ⟩ = 0$ for every $t$ . If $j \neq 0$ this says exactly that every value $r (t)$ lies in the plane through the origin of the relative coordinate with normal $j$ .

2.1step 1.1L1L3L4algebra

Clause 2: assume $j = J e_{3}$ , $J > 0$ . By [L3] there are polar data $(ρ, θ)$ with $r = ρ (\cos θ e_{1} + \sin θ e_{2})$ and $ρ^{2} θ^{'} = J / μ$ , the right side being positive, so the displayed rate identity holds. Since $j = μ r \times r^{'}$ by definition, $r \times r^{'} = j / μ$ ; multiplying by $\frac{1}{2}$ gives $ω = \frac{1}{2} r \times r^{'} = j / (2 μ)$ , a constant vector, and taking norms gives $∣ ω ∣ = J / (2 μ)$ .

3.1step 2.1L3L5algebra

Clause 3: the integrand $s \mapsto ρ (s)^{2} θ^{'} (s)$ is continuous on $I$ (products of continuous functions, [L5]); its primitive $A$ has $A^{'} = \frac{1}{2} ρ^{2} θ^{'} = J / (2 μ)$ by step 2.1, so $A$ is $C^{1}$ . For $t_{0} < t_{1}$ in $I$ , [L5] gives $A (t_{1}) - A (t_{0}) = \int_{t_{0}}^{t_{1}} A^{'} = (t_{1} - t_{0}) J / (2 μ)$ .

4.1step 2.1L1L4algebra∎

Clause 4: by [L1], $x_{1} - X = \frac{m_{2}}{M} r$ and $x_{2} - X = - \frac{m_{1}}{M} r$ ; differentiating with [L4] gives $(x_{1} - X)^{'} = \frac{m_{2}}{M} r^{'}$ and $(x_{2} - X)^{'} = - \frac{m_{1}}{M} r^{'}$ . By bilinearity of the cross product and $\frac{1}{2} r \times r^{'} = ω$ from step 2.1, $\frac{1}{2} (x_{1} - X) \times (x_{1} - X)^{'} = (\frac{m_{2}}{M})^{2} \frac{1}{2} r \times r^{'} = (\frac{m_{2}}{M})^{2} ω,$ and similarly with the sign removed by the two sign changes in the second factor, giving $(\frac{m_{1}}{M})^{2} ω$ ; both vectors are constant because $ω$ is. Their norms are $(\frac{m_{2}}{M})^{2} \frac{J}{2 μ}$ and $(\frac{m_{1}}{M})^{2} \frac{J}{2 μ}$ , whose ratio is $(m_{2} / m_{1})^{2}$ .

Remarks

Reading of the area law. $A$ is the polar sector integral of the reduced orbit; for a simple closed orbit traversed once in the positive polar orientation it is the area enclosed by the trajectory, and clause 3's increment is the standard statement that the radius vector to the force centre sweeps equal areas in equal times. The geometric identification of $A$ with an ordinary plane area for a general (possibly self-intersecting) orbit is not asserted here.
Non-zero angular momentum only. Clauses 2-4 require $j \neq 0$ : for a radial relative motion the nonzero-angular-momentum parametrization is unavailable; a plane containing its constant radial direction and a constant polar angle may still be chosen, the areal-velocity vector vanishes identically, and the equal-area statement is vacuous. The radial case is treated in the scope remark of this page.
Frame and interpretation. The statement is read in one inertial frame; the plane and the constant $j$ are properties of the reduced model in that frame, and neither an observation nor an empirical claim is made.

Physics theoremStatement: AI-adaptedProof: AI-adaptedOpen item page →

Reduction of the central problem to the effective one-dimensional potential: $μ ρ^{''} = - (U_{J}^{e f f})^{'} (ρ)$ , the energy identity, and the converse reconstruction of the planar trajectory from the radial data

Physical scope

The one-dimensional reduction of the non-radial central problem in the adopted model class for angular-momentum magnitude J>0: the radial coordinate takes values in the declared open radial interval J_0 and obeys the effective-potential equation with the centrifugal term, its energy is the constant first integral of that one-dimensional problem, and conversely every C^2 radial solution valued in J_0 together with the angular quadrature reconstructs a planar central-force trajectory with the prescribed angular momentum, hence a motion of the two-label model together with a uniform centre-of-mass motion. The result is a conditional deduction; it asserts no observation, no measured value, and no global existence for either side.

Statement

Adopt the central-force model of Adopted central-force model class: static central pair potentials, the reduced one-label problem, the fixed-centre idealization, reading conventions and exclusions clause (i) with masses $m_{1}, m_{2} > 0$ , reduced mass $μ = m_{1} m_{2} / M$ , $M = m_{1} + m_{2}$ , static central pair potential $u \in C^{2} (J_{0})$ , and let $x : I \to Q$ be a motion on a time interval $I$ with relative coordinate $r$ , attached angular momentum $j = μ r \times r^{'}$ , and suppose $j = J e_{3}$ with $J > 0$ for an oriented orthonormal triad $(e_{1}, e_{2}, e_{3})$ (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, Oriented orthonormal triads in $R^{3}$ : existence over any unit vector, and the orthonormal expansion with no fourth orthogonal direction). Let $(ρ, θ)$ be polar data for $r$ in that triad (Polar reduction of a non-radial central-force trajectory: $ρ^{2} θ^{'} = J / μ$ , the effective-potential equation and energy conservation) and put $U_{J}^{e f f} (ρ) : = u (ρ) + \frac{J^{2}}{2 μ ρ^{2}} (ρ \in J_{0}),$ the effective potential of angular momentum $J$ , of class $C^{2}$ on $J_{0}$ (Central-force trajectories: the central-force equation, angular momentum, energy and the effective potential). Then:

(Radial problem.) On $I$ : $ρ^{2} θ^{'} = J / μ > 0$ , the effective equation $μ ρ^{''} = - (U_{J}^{e f f})^{'} (ρ)$ holds, and the relative energy $E = \frac{1}{2} μ ∥ r^{'} ∥_{2}^{2} + u (∣ r ∣)$ satisfies $E = \frac{1}{2} μ (ρ^{'})^{2} + U_{J}^{e f f} (ρ)$ and is constant on $I$ ; equivalently, the radial coordinate solves the one-dimensional mechanical initial-value problem of mass $μ$ on the declared radial interval $J_{0}$ (an interval contained in $ρ > 0$ ) in the potential $U_{J}^{e f f}$ .
(Reconstruction.) Conversely, let $ρ^{\circ} : I \to R$ be a $C^{2}$ function with $ρ^{\circ} (t) \in J_{0}$ and $μ (ρ^{\circ})^{''} = - (U_{J}^{e f f})^{'} (ρ^{\circ})$ on $I$ , and let $θ^{\circ} : I \to R$ be a $C^{2}$ function with $(θ^{\circ})^{'} = J / (μ (ρ^{\circ})^{2})$ on $I$ (for instance $θ^{\circ} (t) = θ_{0} + \frac{J}{μ} \int_{t_{0}}^{t} ρ^{\circ} (s)^{- 2} d s$ ). Then $r^{\circ} (t) : = ρ^{\circ} (t) (\cos θ^{\circ} (t) e_{1} + \sin θ^{\circ} (t) e_{2}) (t \in I)$ is a central-force trajectory for $(μ, u)$ with $j^{\circ} = μ r^{\circ} \times (r^{\circ})^{'} = J e_{3}$ and constant energy $\frac{1}{2} μ ((ρ^{\circ})^{'})^{2} + U_{J}^{e f f} (ρ^{\circ})$ .
(Mutual inversion and model reconstruction.) The assignments of clauses 1 and 2 are mutually inverse: reconstructing from the polar data of $r$ returns $r$ , and the polar data of the reconstructed $r^{\circ}$ agree with $(ρ^{\circ}, θ^{\circ})$ up to an additive constant of the angular coordinate. Consequently, together with an affine centre-of-mass curve $X^{\circ}$ with $(X^{\circ})^{''} = 0$ on $I$ , the datum $(ρ^{\circ}, θ^{\circ})$ reconstructs a motion of the two-label model with angular momentum $J e_{3}$ , centre of mass $X^{\circ}$ and relative coordinate $r^{\circ}$ (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 clause 4).

Facts & Assumptions

Given: The central-force model of class (i), the motion $x : I \to Q$ on the time interval $I$ , its relative coordinate $r$ , the nonzero constant angular momentum $j = J e_{3}$ , the oriented orthonormal triad, and the polar data $(ρ, θ)$ ; for clause 2, the functions $ρ^{\circ}, θ^{\circ}$ and the curve $r^{\circ}$ of the statement.

[L1]

Reduction and attached objects (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 clauses 3-4, Central-force trajectories: the central-force equation, angular momentum, energy and the effective potential, Adopted central-force model class: static central pair potentials, the reduced one-label problem, the fixed-centre idealization, reading conventions and exclusions clause (ii)): $r$ is a central-force trajectory for $(μ, u)$ on $I$ with $j = μ r \times r^{'} = J e_{3}$ constant and energy $E = \frac{1}{2} μ ∥ r^{'} ∥_{2}^{2} + u (∣ r ∣)$ ; the reduction reconstruction with an affine $X^{\circ}$ is clause 4 of the cited theorem.

[L2]

Polar reduction (Polar reduction of a non-radial central-force trajectory: $ρ^{2} θ^{'} = J / μ$ , the effective-potential equation and energy conservation, Oriented orthonormal triads in $R^{3}$ : existence over any unit vector, and the orthonormal expansion with no fourth orthogonal direction): for a central-force trajectory with constant $j = J e_{3}$ , $J > 0$ , there are twice differentiable $ρ : I \to (0, \infty)$ , $θ : I \to R$ with $r = ρ (\cos θ e_{1} + \sin θ e_{2})$ ; for every such pair, $ρ^{2} θ^{'} = J / μ$ , $μ ρ^{''} = - (U_{J}^{e f f})^{'} (ρ)$ with $U_{J}^{e f f} = u + J^{2} / (2 μ ρ^{2})$ , and $E = \frac{1}{2} μ (ρ^{'})^{2} + U_{J}^{e f f} (ρ)$ is constant; the triad identity $x = \sum_{i} ⟨ x, e_{i} ⟩ e_{i}$ holds.

[L3]

Differentiation rules (Differentiation of sums, scalar multiples, inner products, cross products and squared norms of differentiable curves in $R^{3}$ , The derivatives of sine and cosine are cosine and minus sine, The chain rule, in one line from Carathéodory: if $g$ is differentiable at $c$ and $f$ is differentiable at $g (c)$ , then $f \circ g$ is differentiable at $c$ with $(f \circ g)^{'} (c) = f^{'} (g (c)) g^{'} (c)$ , Sums, scalar multiples, products and quotients: $(f + g)^{'} (c) = f^{'} (c) + g^{'} (c)$ , $(α f)^{'} (c) = α f^{'} (c)$ , $(f g)^{'} (c) = f^{'} (c) g (c) + f (c) g^{'} (c)$ , and $(f / g)^{'} (c) = (f^{'} (c) g (c) - f (c) g^{'} (c)) / g (c)^{2}$ when $g (c) \neq 0$ , Parity and the Pythagorean identity for sine and cosine, $C^{k}$ Euclidean maps and diffeomorphisms, The derivative and the Riemann integral of a vector-valued function: an intrinsic derivative and a componentwise integral, Every continuous function on an interval has a primitive; two primitives differ by a constant; and $\int_{a}^{b} f = G (b) - G (a)$ for any primitive $G$ ): sums, differences, products, quotients with nonvanishing denominators and compositions of $C^{k}$ curves are $C^{k}$ with the usual derivative rules; $(\cos)^{'} = - \sin$ , $(\sin)^{'} = \cos$ , and $\sin^{2} + \cos^{2} = 1$ ; for a continuous $f : I \to R$ on an interval with at least two points, the integral from a fixed base point defines a $C^{1}$ primitive (clause 1 of Every continuous function on an interval has a primitive; two primitives differ by a constant; and $\int_{a}^{b} f = G (b) - G (a)$ for any primitive $G$ ), and any two primitives of $f$ on $I$ differ by a constant (clause 2). The first clause supplies the angular quadrature, and the second supplies angular-phase uniqueness.

[L4]

Cross product (The cross product in $R^{3}$ , The cross product is bilinear, alternating, and orthogonal to both factors, The Euclidean inner product $⟨ x, y ⟩ = \sum_{k < n} x_{k} y_{k}$ on $R^{n}$ ): the cross product is bilinear and alternating, $e_{1} \times e_{2} = e_{3}$ for a positively oriented orthonormal triad, and orthonormality gives $∣ r^{\circ} ∣^{2} = (ρ^{\circ})^{2}$ for the curve of clause 2.

[L5]

The intervals: $I$ has nonempty interior, $J_{0} \subseteq (0, \infty)$ is open, and $U_{J}^{e f f}$ is $C^{2}$ on $J_{0}$ as a sum of $u$ and a quotient with nonvanishing denominator (Intervals of $R$ : the nine order-convex forms, nondegeneracy, and length, Central-force trajectories: the central-force equation, angular momentum, energy and the effective potential, [L3]).

Proof

technique · direct

1.1givenL1L2L5algebra

Clause 1: by [L1], $r$ is a central-force trajectory for $(μ, u)$ with constant $j = J e_{3}$ , so the hypotheses of [L2] hold and its conclusions give $ρ^{2} θ^{'} = J / μ$ , $μ ρ^{''} = - (U_{J}^{e f f})^{'} (ρ)$ , and $E = \frac{1}{2} μ (ρ^{'})^{2} + U_{J}^{e f f} (ρ)$ constant on $I$ ; [L5] makes $U_{J}^{e f f}$ a $C^{2}$ function on $J_{0}$ , so the displayed equation is the one-dimensional mechanical equation for the coordinate $ρ$ on the declared radial interval $J_{0} \subseteq (0, \infty)$ .

1.2givenL3L4algebra

Clause 2, construction and equations: define $n : = \cos θ^{\circ} e_{1} + \sin θ^{\circ} e_{2}$ and $n_{⊥} : = - \sin θ^{\circ} e_{1} + \cos θ^{\circ} e_{2}$ , and $r^{\circ} : = ρ^{\circ} n$ . By [L3], $r^{\circ}$ is $C^{2}$ with $n^{'} = (θ^{\circ})^{'} n_{⊥}$ , $n_{⊥}^{'} = - (θ^{\circ})^{'} n$ and $(r^{\circ})^{'} = (ρ^{\circ})^{'} n + ρ^{\circ} (θ^{\circ})^{'} n_{⊥}$ , hence $(r^{\circ})^{''} = ((ρ^{\circ})^{''} - ρ^{\circ} ((θ^{\circ})^{'})^{2}) n + (2 (ρ^{\circ})^{'} (θ^{\circ})^{'} + ρ^{\circ} (θ^{\circ})^{''}) n_{⊥}$ . With $(θ^{\circ})^{'} = J / (μ (ρ^{\circ})^{2})$ the second coefficient is $\frac{1}{ρ^{\circ}} ((ρ^{\circ})^{2} (θ^{\circ})^{'})^{'} = 0$ , and the radial equation gives $μ ((ρ^{\circ})^{''} - ρ^{\circ} ((θ^{\circ})^{'})^{2}) = μ (ρ^{\circ})^{''} - \frac{J^{2}}{μ (ρ^{\circ})^{3}} = - u^{'} (ρ^{\circ})$ , since $- (U_{J}^{e f f})^{'} (ρ^{\circ}) = - u^{'} (ρ^{\circ}) + \frac{J^{2}}{μ (ρ^{\circ})^{3}}$ . Therefore $μ (r^{\circ})^{''} = - u^{'} (ρ^{\circ}) n = - \frac{u^{'} (∣ r^{\circ} ∣)}{∣ r^{\circ} ∣} r^{\circ}$ , the last equality by $∣ r^{\circ} ∣ = ρ^{\circ}$ and $r^{\circ} = ρ^{\circ} n$ from [L4].

2.1step 1.2L2L3L4algebra

Clause 2, consequences: $∣ r^{\circ} ∣ = ρ^{\circ}$ takes values in $J_{0}$ and $ρ^{\circ}$ is positive, so step 1.2 exhibits $r^{\circ}$ as a central-force trajectory for $(μ, u)$ in the sense of the definition, with $r^{\circ} \times (r^{\circ})^{'} = (ρ^{\circ})^{2} (θ^{\circ})^{'} (n \times n_{⊥}) = (ρ^{\circ})^{2} (θ^{\circ})^{'} e_{3} = \frac{J}{μ} e_{3}$ by [L4] and $n \times n_{⊥} = e_{3}$ ; hence $j^{\circ} = μ r^{\circ} \times (r^{\circ})^{'} = J e_{3}$ . Its energy is $\frac{1}{2} μ ∥ (r^{\circ})^{'} ∥_{2}^{2} + u (ρ^{\circ}) = \frac{1}{2} μ ((ρ^{\circ})^{'})^{2} + \frac{J^{2}}{2 μ (ρ^{\circ})^{2}} + u (ρ^{\circ}) = \frac{1}{2} μ ((ρ^{\circ})^{'})^{2} + U_{J}^{e f f} (ρ^{\circ})$ by [L3] and orthonormality. Since step 1.2 establishes that $r^{\circ}$ is a central-force trajectory with nonzero constant $j^{\circ} = J e_{3}$ , the energy conservation conclusion of [L2] applies and shows that this displayed energy is constant on $I$ .

3.1step 1.1step 1.2step 2.1L1L2L3algebra∎

Clause 3: if $(ρ, θ)$ are the polar data of a trajectory $r$ as in clause 1, the construction of clause 2 applied to $(ρ, θ)$ returns $ρ (\cos θ e_{1} + \sin θ e_{2}) = r$ . Conversely, if $r^{\circ}$ is constructed from $(ρ^{\circ}, θ^{\circ})$ and $(\tilde{ρ}, \tilde{θ})$ is any polar data of $r^{\circ}$ supplied by [L2], then $\tilde{ρ} = ∣ r^{\circ} ∣ = ρ^{\circ}$ and ${\tilde{ρ}}^{2} {\tilde{θ}}^{'} = J / μ = {\tilde{ρ}}^{2} (θ^{\circ})^{'}$ , so $\tilde{θ}$ and $θ^{\circ}$ are primitives of the same continuous function $f (t) = J / (μ (ρ^{\circ} (t))^{2})$ on the nondegenerate interval $I$ . By [L3], clause 2 of Every continuous function on an interval has a primitive; two primitives differ by a constant; and $\int_{a}^{b} f = G (b) - G (a)$ for any primitive $G$ , the two angle functions differ by a constant. Combining with [L1] clause 4, the pair $(ρ^{\circ}, θ^{\circ})$ together with any affine $X^{\circ}$ with $(X^{\circ})^{''} = 0$ on $I$ reconstructs a motion of the two-label model with relative coordinate $r^{\circ}$ and centre of mass $X^{\circ}$ , and by step 2.1 that motion has angular momentum $J e_{3}$ . This completes the proof.

Remarks

The barrier is not a force. $U_{J}^{e f f} = u + J^{2} / (2 μ ρ^{2})$ collects the potential and the centrifugal term of the angular motion; the term is a bookkeeping device of the reduced one-dimensional problem, and no new interaction is introduced.
Energy is the first integral used later. The page's later orbit statements consume the radial equation and the constancy of $E$ from clause 1; clause 2 is what makes the reduction lossless up to the angular phase.
No global existence. Neither clause asserts that a solution of the radial equation exists for all times or that it avoids $ρ \to 0$ ; those are separate questions examined by the subsequent items and the scope remark.

Physics theoremStatement: AI-adaptedProof: AI-adaptedOpen item page →

Circular relative orbits of the central model: the critical-point criterion $u^{'} (r_{0}) = J^{2} / (μ r_{0}^{3})$ , uniform rotation of both labels about the centre of mass, and the linearised radial frequency

Physical scope

Circular relative motion in the adopted central model: the critical-point condition of the effective potential, stated equivalently as the balance $μ r_{0} ω^{2} = u^{'} (r_{0})$ between the central force and the circular motion; the explicit uniform rotation of the relative coordinate and of both labels about the centre of mass; the necessity of the condition for a constant separation; and the frequency of the linearised radial equation at a critical point satisfying the nondegeneracy condition $(U_{J}^{e f f})^{''} (r_{0}) > 0$ . The linearised clause is a statement about the quadratic approximation only; no nonlinear stability, no existence of circular orbits at arbitrary radii and no empirical claim is made.

Statement

(Criterion.) The following are equivalent: $(U_{J}^{e f f})^{'} (r_{0}) = 0; u^{'} (r_{0}) = \frac{J^{2}}{μ r_{0}^{3}}; μ r_{0} ω^{2} = u^{'} (r_{0}) .$
(Existence and the two labels.) If the condition of clause 1 holds, then for every $θ_{0} \in R$ the curve $r^{\circ} (t) : = r_{0} (\cos (θ_{0} + ω t) e_{1} + \sin (θ_{0} + ω t) e_{2}) (t \in R)$ is a central-force trajectory for $(μ, u)$ defined on all of $R$ , with $∣ r^{\circ} ∣ \equiv r_{0}$ , angular momentum $j = μ r^{\circ} \times (r^{\circ})^{'} = J e_{3}$ and constant energy $E = U_{J}^{e f f} (r_{0})$ ; and for every interval $I \subseteq I_{0}$ with nonempty interior and affine centre-of-mass curve $X^{\circ} : I \to R^{3}$ with $(X^{\circ})^{''} = 0$ , the restrictions to $I$ reconstruct a motion of the two-label model (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 clause 4) has both labels moving uniformly on circles about the centre of mass: $∣ x_{1} (t) - X^{\circ} (t) ∣ = \frac{m_{2}}{M} r_{0}, ∣ x_{2} (t) - X^{\circ} (t) ∣ = \frac{m_{1}}{M} r_{0},$ with angular velocity $ω$ about $X^{\circ}$ .
(Necessity.) Conversely, if a motion of the model has constant separation $∣ r ∣ \equiv r_{0}$ on a time interval $I$ with nonempty interior and nonzero constant angular momentum $j = J e_{3}$ , then $(U_{J}^{e f f})^{'} (r_{0}) = 0$ and its polar angular coordinate satisfies $θ^{'} \equiv ω$ ; the relative motion is the uniform rotation of clause 2 up to the initial value of $θ$ , and both labels move uniformly on the circles of clause 2 about the affine centre of mass.
(Linearised radial frequency.) If $(U_{J}^{e f f})^{'} (r_{0}) = 0$ and $(U_{J}^{e f f})^{''} (r_{0}) > 0$ , then $ω_{l i n} : = \sqrt{(U_{J}^{e f f})^{''} (r_{0}) / μ} > 0$ , and a twice differentiable $δ : I \to R$ on an interval $I$ with at least two points solves the linearised radial equation $μ δ^{''} = - (U_{J}^{e f f})^{''} (r_{0}) δ on I$ if and only if $δ (t) = A \cos (ω_{l i n} (t - t_{0})) + B \sin (ω_{l i n} (t - t_{0}))$ for some $A, B \in R$ and $t_{0} \in I$ . This is a statement about the quadratic radial approximation at $r_{0}$ ; it does not assert nonlinear stability of the circular orbit.

Facts & Assumptions

Given: The central-force model of class (i) with the $C^{2}$ potential $u$ on the open interval $J_{0} \subseteq (0, \infty)$ , the data $J > 0$ , the oriented orthonormal triad $(e_{1}, e_{2}, e_{3})$ , the radius $r_{0} \in J_{0}$ , and the functions $U_{J}^{e f f}$ and constant $ω$ of the statement.

[L1]

Reduction and attached objects (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 clauses 3-4, Central-force trajectories: the central-force equation, angular momentum, energy and the effective potential, Adopted central-force model class: static central pair potentials, the reduced one-label problem, the fixed-centre idealization, reading conventions and exclusions clause (ii)): the relative coordinate of every motion is a central-force trajectory for $(μ, u)$ ; conversely every central-force trajectory for $(μ, u)$ together with an affine centre-of-mass curve reconstructs a motion of the model, with $x_{1} - X^{\circ} = \frac{m_{2}}{M} r$ and $x_{2} - X^{\circ} = - \frac{m_{1}}{M} r$ .

[L2]

Mathematical supplier (Circular orbits of a non-radial central-force trajectory: the critical-point condition and the squared frequency of the linearised radial equation clauses 1-3, Oriented orthonormal triads in $R^{3}$ : existence over any unit vector, and the orthonormal expansion with no fourth orthogonal direction, Solutions of $v^{''} + ω^{2} v = 0$ are exactly the functions $A \cos (ω (t - t_{0})) + B \sin (ω (t - t_{0}))$ , Central-force trajectories: the central-force equation, angular momentum, energy and the effective potential): for a central-force trajectory for $(μ, u)$ with constant $j = J e_{3}$ , $J > 0$ , and $r_{0} \in J_{0}$ , (i) if $ρ \equiv r_{0}$ then $(U_{J}^{e f f})^{'} (r_{0}) = 0$ and $θ^{'} \equiv J / (μ r_{0}^{2})$ , the trajectory being $r_{0} (\cos θ e_{1} + \sin θ e_{2})$ with $θ (t) = θ_{0} + ω t$ ; (ii) if $(U_{J}^{e f f})^{'} (r_{0}) = 0$ then for every $θ_{0}$ the displayed $r^{\circ}$ is a central-force trajectory on $R$ with $j = J e_{3}$ and $E = U_{J}^{e f f} (r_{0})$ ; (iii) if $(U_{J}^{e f f})^{'} (r_{0}) = 0$ and $(U_{J}^{e f f})^{''} (r_{0}) > 0$ , the solutions of the linearised equation $μ δ^{''} = - (U_{J}^{e f f})^{''} (r_{0}) δ$ on an interval with at least two points are exactly $A \cos (ω_{l i n} (t - t_{0})) + B \sin (ω_{l i n} (t - t_{0}))$ with $ω_{l i n} = \sqrt{(U_{J}^{e f f})^{''} (r_{0}) / μ}$ .

[L3]

Differentiation and trigonometry (The chain rule, in one line from Carathéodory: if $g$ is differentiable at $c$ and $f$ is differentiable at $g (c)$ , then $f \circ g$ is differentiable at $c$ with $(f \circ g)^{'} (c) = f^{'} (g (c)) g^{'} (c)$ , Sums, scalar multiples, products and quotients: $(f + g)^{'} (c) = f^{'} (c) + g^{'} (c)$ , $(α f)^{'} (c) = α f^{'} (c)$ , $(f g)^{'} (c) = f^{'} (c) g (c) + f (c) g^{'} (c)$ , and $(f / g)^{'} (c) = (f^{'} (c) g (c) - f (c) g^{'} (c)) / g (c)^{2}$ when $g (c) \neq 0$ , $C^{k}$ Euclidean maps and diffeomorphisms): derivatives of the displayed trigonometric curves and of the potential expressions are computed by the chain rule and the algebra of derivatives; the barrier term $ρ \mapsto J^{2} / (2 μ ρ^{2})$ has derivative $- J^{2} / (μ ρ^{3})$ on $(0, \infty)$ .

[L4]

Intervals and dimensions: $J_{0} \subseteq (0, \infty)$ is open, $r_{0} > 0$ , and $J > 0$ (Intervals of $R$ : the nine order-convex forms, nondegeneracy, and length).

Proof

technique · direct

1.1givenL3L4algebra

Clause 1: differentiating the definition of $U_{J}^{e f f}$ with [L3] gives $(U_{J}^{e f f})^{'} (r_{0}) = u^{'} (r_{0}) - J^{2} / (μ r_{0}^{3})$ , so the first two conditions are equivalent; and $μ r_{0} ω^{2} = μ r_{0} \cdot J^{2} / (μ^{2} r_{0}^{4}) = J^{2} / (μ r_{0}^{3})$ by the definition of $ω$ , so $u^{'} (r_{0}) = J^{2} / (μ r_{0}^{3})$ is equivalent to $μ r_{0} ω^{2} = u^{'} (r_{0})$ .

2.1step 1.1L1L2algebra

Clause 2: assume $(U_{J}^{e f f})^{'} (r_{0}) = 0$ . By [L2] clause (ii) the displayed curve $r^{\circ}$ is a central-force trajectory for $(μ, u)$ on $R$ with $∣ r^{\circ} ∣ \equiv r_{0}$ , $j = J e_{3}$ and $E = U_{J}^{e f f} (r_{0})$ ; restricted to any interval $I \subseteq I_{0}$ with nonempty interior and combined with an affine $X^{\circ} : I \to R^{3}$ , [L1] clause 4 reconstructs a motion on $I$ of the two-label model with $x_{1} - X^{\circ} = \frac{m_{2}}{M} r^{\circ}$ and $x_{2} - X^{\circ} = - \frac{m_{1}}{M} r^{\circ}$ . Taking norms gives the displayed constant radii; and since $(r^{\circ})^{'}$ is tangent to the circle $∣ y ∣ = r_{0}$ and $∣ (r^{\circ})^{'} ∣ = r_{0} ω$ , each label traverses its circle with angular velocity $ω$ .

2.2step 1.1L1L2algebra

Clause 3: assume $∣ r ∣ \equiv r_{0}$ on $I$ with $j = J e_{3}$ , $J > 0$ . By [L1] and [L2] clause (i), $(U_{J}^{e f f})^{'} (r_{0}) = 0$ , the polar angular coordinate satisfies $θ^{'} \equiv J / (μ r_{0}^{2}) = ω$ , so $θ (t) = θ_{0} + ω t$ on $I$ up to the additive constant, and $r = r_{0} (\cos θ e_{1} + \sin θ e_{2})$ is the uniform rotation of clause 2; the reconstruction of [L1] clause 4 gives the two labels on the circles of clause 2 about the affine centre of mass.

3.1step 1.1step 2.1L2L4algebra∎

Clause 4: under the two derivatives hypotheses, [L2] clause (iii) gives exactly the stated solutions of the linearised equation with the stated positive frequency $ω_{l i n} = \sqrt{(U_{J}^{e f f})^{''} (r_{0}) / μ}$ , as an identity on any interval with at least two points containing $t_{0}$ .

Remarks

The criterion is a force balance. The condition $μ r_{0} ω^{2} = u^{'} (r_{0})$ says that the effective central force at $r_{0}$ provides exactly the centripetal acceleration of uniform circular motion with angular velocity $ω$ ; no circular orbit exists at a radius where the effective potential has no critical point.
Linearisation only. Clause 4 is a statement about the quadratic approximation of the radial equation. Nothing here proves that small radial perturbations of the circular orbit stay small for all future times; only a nonlinear argument could, and none is made on this page.
Model status. $J$ , $r_{0}$ , $u$ and the masses are model data. The statement is conditional on the adopted central model and reports no observation.

PostulateStatement: AI-adaptedProof: Not applicableOpen item page →

Adopted two-label gravitational model: the inverse-square pair potential with coupling parameters, the empirical mass identification, and the measured value and uncertainty of the Newtonian constant

Physical scope

Adoption of the two-label gravitational model inside the page's central-force class: the inverse-square static pair potential with the Newtonian constant and positive gravitational coupling parameters; the empirical identification of the coupling parameters with the inertial masses, adopted as an assumption of the page's applications; the measured value of the Newtonian constant with its uncertainty and the mutual inconsistency of its measurements; and the non-relativistic weak-field two-point-label applicability regime, specified by the source-potential and individual-speed inequalities in a specified comparison frame in clause (iv), with its exclusions. The item adopts model data and empirical premises; it asserts no law of nature, no proof of the identification, and no numerical prediction by itself.

Postulate

Adopt the following gravitational model and empirical premises for the two-body part of this page, 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 and the central-force class of Adopted central-force model class: static central pair potentials, the reduced one-label problem, the fixed-centre idealization, reading conventions and exclusions clause (i).

(i) The two-label gravitational model. A model of the gravitational branch is a model of class (i) of Adopted central-force model class: static central pair potentials, the reduced one-label problem, the fixed-centre idealization, reading conventions and exclusions with $J_{0} = (0, \infty)$ , $Q = {x = (x_{1}, x_{2}) : x_{1} \neq x_{2}}$ , masses $m_{1}, m_{2} > 0$ in the equation of motion, and the gravitational pair potential $U (x_{1}, x_{2}) = - \frac{G_{N} m_{1}^{g} m_{2}^{g}}{∣ x_{1} - x_{2} ∣},$ where $G_{N} > 0$ is the Newtonian constant of gravitation, a constant of dimension $ℓ^{3} μ^{- 1} τ^{- 2}$ read in $m^{3} k g^{- 1} s^{- 2}$ , and $m_{1}^{g}, m_{2}^{g} > 0$ are the gravitational coupling parameters of the labels, primitive model data distinct from the inertial masses $m_{1}, m_{2}$ (Restricted interaction models: pair decompositions, antisymmetry and centrality options, potential forces and the gravitational branch clause (v), Dimensions and SI unit conventions of the non-relativistic classical mechanics framework). The corresponding central pair potential of class (i) is $u (ρ) = - κ / ρ$ with $κ : = G_{N} m_{1}^{g} m_{2}^{g} > 0, dimension μ ℓ^{3} τ^{- 2};$ it is of class $C^{\infty}$ on $(0, \infty)$ , the model has no external forces, and coincident configurations are excluded. The force is attractive, and its inverse-square form is an adopted law of this branch, not a consequence of the equation of motion.

(ii) The empirical mass identification. For the quantitative applications of this page, adopt in addition the identification $m_{i}^{g} = m_{i} (i = 1, 2)$ of the gravitational coupling parameters with the inertial masses. This is an empirical assumption, not part of clause (i) and not a consequence of the equation of motion (Restricted interaction models: pair decompositions, antisymmetry and centrality options, potential forces and the gravitational branch clause (vi)); it is the content of the weak equivalence principle for these labels, and it is adopted here only where a statement explicitly says so. When it is adopted, $κ = G_{N} m_{1} m_{2}$ , and every conclusion that uses it inherits its empirical status; statements that never invoke clause (ii) remain conditional on clause (i) alone, with $κ$ expressed through $m_{1}^{g}, m_{2}^{g}$ .

(iii) The measured value of the constant. When a model of this page is used quantitatively, $G_{N}$ may be assigned the CODATA 2022 recommended value $G_{N} = 6.674 30 (15) \times 10^{- 11} m^{3} k g^{- 1} s^{- 2},$ with standard uncertainty $0.000 15 \times 10^{- 11}$ in the same unit and relative standard uncertainty $2.2 \times 10^{- 5}$ : a qualified empirical premise, not a definition and not an exact value. The recommendation is a least-squares value from 16 measurements whose mutual inconsistency led the CODATA Task Group to expand its uncertainty by a factor of $3.9$ , and the 2022 value agrees with the 2018 value because the input data were unchanged. Any numerical conclusion that uses this assignment inherits the uncertainty of $G_{N}$ together with the uncertainties and conditions of the other model data actually used; no such numerical conclusion is drawn by this postulate itself.

(iv) Scope and exclusions. The force equations in clause (i) define the formal Newtonian point-particle model. The words weak-field and non-relativistic specify when it is intended as an approximation to a relativistic gravitational description. Let $c : = 299792458 m s^{- 1}$ be the speed of light in vacuum, used here only as an external comparison constant, not as a parameter in the Newtonian equations. For a configuration $x = (x_{1}, x_{2}) \in Q$ with $ρ (x) : = ∣ x_{1} - x_{2} ∣ > 0$ , define the source potentials per unit gravitational coupling as scalar functions $Φ_{1}, Φ_{2} : Q \to R$ by $Φ_{1} (x) : = - \frac{G_{N} m_{2}^{g}}{ρ (x)}, Φ_{2} (x) : = - \frac{G_{N} m_{1}^{g}}{ρ (x)},$ each of dimension $ℓ^{2} τ^{- 2}$ . Fix the inertial frame of Adopted force model: force maps, the equation of motion, additive composition of force contributions and frame covariance clause (i), with its time and spatial coordinates, as the comparison frame for the intended relativistic approximation. For a state velocity $v = (v_{1}, v_{2}) \in (R^{3})^{2}$ , where $v_{i}$ is the coordinate velocity of label $i$ in that frame, define the dimensionless parameters $ϵ_{Φ} (x) : = \frac{\max {∣ Φ_{1} (x) ∣, ∣ Φ_{2} (x) ∣}}{c^{2}}, ϵ_{v} (v) : = \frac{\max {∣ v_{1} ∣^{2}, ∣ v_{2} ∣^{2}}}{c^{2}} .$ For a motion $x : J \to Q$ , a trajectory interval $I \subseteq J$ and a dimensionless tolerance $η \in (0, 1)$ chosen small enough for the intended approximation, the weak-field, slow-motion regime in this comparison frame on that segment means $ϵ_{Φ} (x (t)) \leq η and ϵ_{v} (x^{'} (t)) \leq η for every t \in I .$ Small relative speed alone does not meet the speed condition: adding a common large velocity leaves $v_{1} - v_{2}$ unchanged. The formal Newtonian equations retain their frame transformation rule, but this approximation criterion is assessed in the specified comparison coordinates, as in the cited Newtonian-limit discussion; it is not asserted in every boosted frame. The choice of $η$ is application-specific; these conditions are applicability criteria, not a proved sufficient condition for a relativistic error bound. This model supplies no relativistic correction estimate or error bound. When $G_{N}$ is assigned the value in clause (iii), these inequalities inherit its standard uncertainty and any uncertainty in the other inputs; no regime assessment is made here. The model treats the two labels as point particles with fixed masses, ignores external bodies, rotation, deformation, mass loss, radiation and any non-gravitational force, and excludes coincident configurations. It is not asserted that any physical system is exactly of this form; in particular, the identification of clause (ii) is not a framework-wide identification, and the inverse-square law is not derived from the balance laws of the framework.

(v) Status. Clauses (i), (ii) and (iii) are adopted assumptions of the model and its empirical inputs; this item contains no proof and no measurement of its own. The theorems of this page that invoke the model are conditional on these clauses, and agreement of their predictions with observations would test the combined model and apparatus assumptions rather than prove them.

Status and scope

What is primitive and what is derived. The inertial masses, the gravitational couplings, the constant $G_{N}$ and the pair form are model data; the reduced mass $μ = m_{1} m_{2} / (m_{1} + m_{2})$ is derived from the inertial masses, and $κ = G_{N} m_{1}^{g} m_{2}^{g}$ (or $G_{N} m_{1} m_{2}$ under clause (ii)) is the parameter that enters the central-force statements.
Why the identification is separated. Momentum, angular momentum and energy balances and the relative reduction use only the inertial masses and the potential; the equality of gravitational and inertial parameters enters only through clause (ii), so its empirical content is not silently absorbed into the mechanical theorems.
Why the constant is separated. $G_{N}$ is measured, not exact; a page that replaces it by a number must say so and must carry the uncertainty. The symbolic theorems of this page use the symbol; only explicitly numerical applications would use clause (iii).

Physics theoremStatement: AI-adaptedProof: AI-adaptedOpen item page →

The inverse-square gravitational relative trajectory has the Kepler polar form: $ρ = p / (1 + e \cos (θ - θ_{0}))$ , $E = (μ κ^{2} / 2 J^{2}) (e^{2} - 1)$ , elliptic completeness and barycentric scaled ellipses

Physical scope

The Keplerian orbit classification of the two-label gravitational model with non-zero angular momentum: the polar conic with its eccentricity and focus at the force centre, the eccentricity-energy relation and the elliptic, parabolic and hyperbolic branches, the centrifugal barrier that keeps the motion away from the collision, and - in the elliptic case - the completeness of the relative trajectory and reconstruction on the declared model time window, the closed periodic relative orbit with the ellipse geometry, and the scaled barycentric conics of both labels. The parabolic and hyperbolic clauses record only the orbit formula, the periapsis and the domain of the angle on the actual domain of the motion; no asymptotic extent, scattering or observational claim is made.

Statement

Assume the Axiom of Countable Choice (The Axiom of Countable Choice ( ${A C}_{ω}$ )). Adopt the two-label gravitational model of Adopted two-label gravitational model: the inverse-square pair potential with coupling parameters, the empirical mass identification, and the measured value and uncertainty of the Newtonian constant clause (i): inertial masses $m_{1}, m_{2} > 0$ , $M = m_{1} + m_{2}$ , reduced mass $μ = m_{1} m_{2} / M$ , gravitational couplings $m_{1}^{g}, m_{2}^{g} > 0$ , $κ = G_{N} m_{1}^{g} m_{2}^{g} > 0$ , and the central pair potential $u (ρ) = - κ / ρ$ on $J_{0} = (0, \infty)$ , and model time window $I_{0}$ . Let $x : I \to Q$ be a motion on a time interval $I$ with nonempty interior, with relative coordinate $r$ , attached angular momentum $j = μ r \times r^{'}$ and energy $E = \frac{1}{2} μ ∥ r^{'} ∥_{2}^{2} - κ / ∣ r ∣$ (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, Central-force trajectories: the central-force equation, angular momentum, energy and the effective potential), and suppose $j = J e_{3}$ , $J > 0$ , for an oriented orthonormal triad $(e_{1}, e_{2}, e_{3})$ . Let $(ρ, θ)$ be polar data for $r$ and put $p : = \frac{J^{2}}{μ κ}, U_{J}^{e f f} (ρ) : = - \frac{κ}{ρ} + \frac{J^{2}}{2 μ ρ^{2}}, ρ_{*} : = \frac{J^{2}}{μ κ} = p .$ Then:

In the orbital plane $Π : = span (e_{1}, e_{2})$ , for a point $F \in Π$ , a line $D \subseteq Π$ with $F \notin D$ , and $e > 0$ , define the focus-directrix conic of eccentricity $e$ to be $C (F, D, e) : = {y \in Π : ∥ y - F ∥_{2} = e d_{Π} (y, D)}, d_{Π} (y, D) : = \inf_{z \in D} ∥ y - z ∥_{2} .$ Here $F$ is a focus, $D$ is its directrix, and $e$ is the distance ratio in the definition. The types $0 < e < 1$ , $e = 1$ , and $e > 1$ are called ellipse, parabola, and hyperbola, respectively. The case $e = 0$ is the circle of radius $p$ centered at $F$ , the circular ellipse with coincident foci at its centre. In this item, an unbounded conic is a conic locus that is unbounded, not an open subset of $Π$ ; the periapsis distance is the minimum distance on the conic to its focus, and any point attaining it is a periapsis.

(Conic and eccentricity.) There are $e \geq 0$ and $θ_{0} \in R$ with $ρ = \frac{p}{1 + e \cos (θ - θ_{0})}$ on the interior of $I$ , and the energy satisfies $E = \frac{μ κ^{2}}{2 J^{2}} (e^{2} - 1)$ ; consequently $E < 0 ⟺ e < 1$ , $E = 0 ⟺ e = 1$ and $E > 0 ⟺ e > 1$ . The relative trajectory lies on a focus-directrix conic with the force centre as focus when $e > 0$ : its directrix is $D = {y \in Π : ⟨ y, n_{0} ⟩ = p / e}$ , where $n_{0} : = \cos θ_{0} e_{1} + \sin θ_{0} e_{2}$ . When $e = 0$ , the relative trajectory lies on the circle of radius $p$ centered at the force centre.
(Centrifugal barrier.) $ρ_{*}$ is the unique critical point of $U_{J}^{e f f}$ in $(0, \infty)$ and the unique minimum, with $U_{J}^{e f f} (ρ_{*}) = - \frac{μ κ^{2}}{2 J^{2}};$ setting $ρ_{\min} (E) : = \frac{J^{2}}{μ κ + \sqrt{μ^{2} κ^{2} + 2 μ E J^{2}}} > 0,$ one has $U_{J}^{e f f} (ρ_{\min} (E)) = E$ and $ρ (t) \geq ρ_{\min} (E)$ for every $t \in I$ . In particular the relative motion never reaches the collision $r = 0$ , and $ρ_{\min} (E) = p / (1 + e)$ is the periapsis distance of the conic of clause 1.
(Elliptic case.) If $E < 0$ , then $e < 1$ and: (a) the relative trajectory extends uniquely to a central-force trajectory for $(μ, u)$ defined on all of $R$ . Together with the unique affine extension of $X$ , this defines global coordinate curves ${\overset{ˉ}{x}}_{1} = \overset{ˉ}{X} + (m_{2} / M) \overset{ˉ}{r}$ , ${\overset{ˉ}{x}}_{2} = \overset{ˉ}{X} - (m_{1} / M) \overset{ˉ}{r}$ satisfying the same time-independent force equations; their restriction to $I_{0}$ is the unique extension of the model motion $x$ . They are a model motion on all of $R$ when $I_{0} = R$ . The relative motion is periodic, with $θ (t + T) = θ (t) + 2 π$ for one full angular revolution; (b) $r$ is an ellipse with focus at the force centre, semiaxes $a = \frac{p}{1 - e^{2}} = - \frac{κ}{2 E}, b = a \sqrt{1 - e^{2}},$ and $p / (1 + e) \leq ρ \leq p / (1 - e)$ ; (c) each label's trajectory relative to the centre of mass is the conic of clause 1 scaled by $m_{j} / M$ : with $p_{1} : = \frac{m_{2}}{M} p$ , $p_{2} : = \frac{m_{1}}{M} p$ one has $∣ x_{1} - X ∣ = \frac{p_{1}}{1 + e \cos (θ - θ_{0})}, ∣ x_{2} - X ∣ = \frac{p_{2}}{1 + e \cos (θ - θ_{0})},$ so both labels trace ellipses with the centre of mass at a focus, the same eccentricity $e$ and semimajor axes $a_{1} = \frac{m_{2}}{M} a$ , $a_{2} = \frac{m_{1}}{M} a$ .
(Parabolic and hyperbolic cases on the domain of the motion.) If $E = 0$ , then $e = 1$ and $ρ = p / (1 + \cos (θ - θ_{0}))$ with periapsis $ρ_{\min} (0) = p / 2$ . If $E > 0$ , then $e > 1$ and the formula of clause 1 holds at every $t$ with $1 + e \cos (θ - θ_{0}) > 0$ , with periapsis $ρ_{\min} (E) = p / (1 + e)$ . In both cases $e \geq 1$ and the corresponding conic locus is unbounded; the extent of the trajectory, its asymptotes and its scattering interpretation are not claimed here.

Facts & Assumptions

Given: The gravitational two-label model with the data $μ, κ, J > 0$ , the motion $x : I \to Q$ on the time interval $I$ , the relative coordinate $r$ , the polar data $(ρ, θ)$ , and the constants $p$ , $ρ_{*}$ , $U_{J}^{e f f}$ of the statement.

[A1]

The Axiom of Countable Choice is assumed (The Axiom of Countable Choice ( ${A C}_{ω}$ )).

[L1]

Reduction and classification input (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 clauses 3-4, Central-force trajectories: the central-force equation, angular momentum, energy and the effective potential, Adopted central-force model class: static central pair potentials, the reduced one-label problem, the fixed-centre idealization, reading conventions and exclusions clauses (i)-(ii), Adopted two-label gravitational model: the inverse-square pair potential with coupling parameters, the empirical mass identification, and the measured value and uncertainty of the Newtonian constant clause (i)): $r$ is a central-force trajectory for $(μ, u)$ with $∣ r (t) ∣ \in (0, \infty)$ , constant $j = μ r \times r^{'} = J e_{3}$ and constant energy $E$ ; $x_{1} - X = \frac{m_{2}}{M} r$ and $x_{2} - X = - \frac{m_{1}}{M} r$ with $X$ affine.

[L2]

Mathematical orbit theory (Binet's equation for the attractive inverse-square field: $d^{2} u / d φ^{2} + u = μ κ / J^{2}$ for $u = 1 / ρ$ , The Kepler orbit equation is a conic: $u = c (1 + e \cos (φ - θ_{0}))$ with $E = (μ κ^{2} / 2 J^{2}) (e^{2} - 1)$ , The subcritical Kepler conic is an ellipse with semiaxes $a = p / (1 - e^{2})$ , $b = a \sqrt{1 - e^{2}}$ , focus at the origin and area $π a b$ ; its area is also $\frac{1}{2} \int_{0}^{2 π} ϱ^{2}$ , Completeness and periodicity of the negative-energy Kepler orbit, with $T = 2 π μ a b / J$ and $T^{2} = 4 π^{2} μ a^{3} / κ$ , Polar reduction of a non-radial central-force trajectory: $ρ^{2} θ^{'} = J / μ$ , the effective-potential equation and energy conservation, Oriented orthonormal triads in $R^{3}$ : existence over any unit vector, and the orthonormal expansion with no fourth orthogonal direction): for the attractive inverse-square central-force trajectory with $j = J e_{3}$ , $J > 0$ , there are polar data with $ρ^{2} θ^{'} = J / μ$ and a conic representation $ρ = p / (1 + e \cos (θ - θ_{0}))$ on $I^{\circ}$ , where $p = J^{2} / (μ κ)$ , $E = (μ κ^{2} / 2 J^{2}) (e^{2} - 1)$ and the sign classification hold; if $E < 0$ then the trajectory extends uniquely to all of $R$ with $ρ$ bounded in $[p / (1 + e), p / (1 - e)]$ and is periodic with $θ (t + T) = θ (t) + 2 π$ ; and the subcritical conic is an ellipse with semiaxes $a = p / (1 - e^{2})$ , $b = a \sqrt{1 - e^{2}}$ , focus at the origin, and area $π a b$ .

[L3]

Elementary computations (The chain rule, in one line from Carathéodory: if $g$ is differentiable at $c$ and $f$ is differentiable at $g (c)$ , then $f \circ g$ is differentiable at $c$ with $(f \circ g)^{'} (c) = f^{'} (g (c)) g^{'} (c)$ , Sums, scalar multiples, products and quotients: $(f + g)^{'} (c) = f^{'} (c) + g^{'} (c)$ , $(α f)^{'} (c) = α f^{'} (c)$ , $(f g)^{'} (c) = f^{'} (c) g (c) + f (c) g^{'} (c)$ , and $(f / g)^{'} (c) = (f^{'} (c) g (c) - f (c) g^{'} (c)) / g (c)^{2}$ when $g (c) \neq 0$ , $C^{k}$ Euclidean maps and diffeomorphisms, The Euclidean inner product $⟨ x, y ⟩ = \sum_{k < n} x_{k} y_{k}$ on $R^{n}$ , Intervals of $R$ : the nine order-convex forms, nondegeneracy, and length): derivatives and algebraic identities of the displayed rational and trigonometric expressions; $J, μ, κ > 0$ and $E \geq U_{J}^{e f f} (ρ_{*})$ .

Proof

technique · direct

1.1givenA1L1L2algebra

Clause 1: by [L1] the curve $r$ is a central-force trajectory for $(μ, u)$ with constant nonzero angular momentum $J e_{3}$ and constant energy $E$ , so the hypotheses of the inverse-square orbit theory [L2] are met and its polar representation and eccentricity-energy relation give the displayed formula and sign equivalences, since $μ κ^{2} / (2 J^{2}) > 0$ . To verify the geometric description, for $e > 0$ put $n_{0} : = \cos θ_{0} e_{1} + \sin θ_{0} e_{2}$ and $D : = {y \in Π : ⟨ y, n_{0} ⟩ = p / e}$ . For $t \in I^{\circ}$ , set $ϕ : = θ (t) - θ_{0}$ . The formula and its positive denominator give $ρ (t) \cos ϕ < p / e$ and $e d_{Π} (r (t), D) = e (p / e - ρ (t) \cos ϕ) = p - e ρ (t) \cos ϕ = ρ (t) = ∥ r (t) ∥_{2}$ , so $r (t) \in C (0, D, e)$ and the force centre is a focus. If $e = 0$ , the polar formula gives $ρ (t) = p$ , so $r (t)$ lies on the stated circle.

2.1step 1.1L2L3algebra

Clause 2: differentiating $U_{J}^{e f f}$ gives $(U_{J}^{e f f})^{'} (ρ) = (κ ρ - J^{2} / μ) / ρ^{3}$ , which vanishes exactly at $ρ_{*}$ and is negative for $ρ < ρ_{*}$ and positive for $ρ > ρ_{*}$ ; for $0 < ρ < ρ^{'} \leq ρ_{*}$ the difference identity $U_{J}^{e f f} (ρ^{'}) - U_{J}^{e f f} (ρ) = \frac{(ρ^{'} - ρ) [2 μ κ ρ ρ^{'} - J^{2} (ρ^{'} + ρ)]}{2 μ ρ^{2} (ρ^{'})^{2}}$ has numerator at most $(ρ^{'} - ρ) [2 J^{2} ρ - J^{2} (ρ^{'} + ρ)] = - J^{2} (ρ^{'} - ρ)^{2} < 0$ , where the inequality uses $ρ \leq ρ^{'}$ and $ρ^{'} \leq ρ_{*} = J^{2} / (μ κ)$ ; hence $U_{J}^{e f f}$ is strictly decreasing on $(0, ρ_{*}]$ and $ρ_{*}$ is its unique minimum. Direct substitution gives $- \frac{κ}{ρ_{\min} (E)} + \frac{J^{2}}{2 μ ρ_{\min} (E)^{2}} = E,$ by multiplying by $ρ_{\min}^{}$ , clearing denominators with the definition of $ρ_{\min}$ and using $(μ κ + \sqrt{μ^{2} κ^{2} + 2 μ E J^{2}}) (- μ κ + \sqrt{μ^{2} κ^{2} + 2 μ E J^{2}}) = 2 μ E J^{2}$ . Along the motion, the energy identity of [L2] gives $\frac{1}{2} μ (ρ^{'})^{2} + U_{J}^{e f f} (ρ) = E$ , so $U_{J}^{e f f} (ρ (t)) \leq E = U_{J}^{e f f} (ρ_{\min} (E))$ ; if $ρ (t) \leq ρ_{*}$ then strict decrease on $(0, ρ_{*}]$ forces $ρ (t) \geq ρ_{\min} (E)$ , while if $ρ (t) > ρ_{*}$ then $ρ (t) > ρ_{*} \geq ρ_{\min} (E)$ because $ρ_{\min} (E) \leq ρ_{*}$ (its defining denominator is at least $μ κ$ ). Hence $ρ \geq ρ_{\min} (E) > 0$ throughout.

3.1step 1.1step 2.1L1L2algebra

Clause 3: assume $E < 0$ , so $e < 1$ by step 1.1. Clause (a) is [L2] for the global relative trajectory $\overset{ˉ}{r}$ . The affine curve $X$ has a unique affine extension $\overset{ˉ}{X}$ to $R$ because $I$ has at least two points. Define ${\overset{ˉ}{x}}_{1} = \overset{ˉ}{X} + (m_{2} / M) \overset{ˉ}{r}$ and ${\overset{ˉ}{x}}_{2} = \overset{ˉ}{X} - (m_{1} / M) \overset{ˉ}{r}$ globally; the calculation in [L1] clause 4 shows that these satisfy the same time-independent equations. Their restriction to $I_{0}$ is a motion of the original model by that clause, and uniqueness follows from the unique relative and affine extensions. A global coordinate curve is a model motion on all of $R$ only when the model time window is $I_{0} = R$ . Clause (b) is the ellipse clause of [L2], including $a = p / (1 - e^{2}) = - κ / (2 E)$ and the bound on $ρ$ . Clause (c): by [L1] and [L2], $∣ x_{1} - X ∣ = \frac{m_{2}}{M} ∣ r ∣ = \frac{m_{2}}{M} ρ$ and $∣ x_{2} - X ∣ = \frac{m_{1}}{M} ρ$ , so substituting the conic of clause 1 gives the two displayed conic formulas; their semimajor axes are $p_{j} / (1 - e^{2}) = \frac{m_{j}}{M} a$ with the same eccentricity and the focus at the origin of $x_{i} - X$ , which is the centre of mass.

4.1step 1.1step 2.1L1L2algebra∎

Clause 4 and endpoints: [L2] gives $ρ = p / (1 + e \cos (θ - θ_{0}))$ on $I^{\circ}$ . Equivalently, the continuous functions $t \mapsto ρ (t) (1 + e \cos (θ (t) - θ_{0}))$ and $t \mapsto p$ agree on $I^{\circ}$ , so they agree on its closure in $I$ , which is all of $I$ ; since $ρ (t) > 0$ and $p > 0$ , this identity gives $1 + e \cos (θ (t) - θ_{0}) = p / ρ (t) > 0$ and the conic formula also holds at every included endpoint of $I$ . If $E = 0$ then $e = 1$ by step 1.1; the periapsis is $p / 2 = ρ_{\min} (0)$ by step 2.1. The conic locus is unbounded because $1 + \cos ϕ ↓ 0$ as $ϕ ↑ π$ through values below $π$ . If $E > 0$ then $e > 1$ ; the periapsis is $ρ_{\min} (E) = p / (1 + e)$ by step 2.1. With $ϕ_{*} : = \arccos (- 1 / e) \in (0, π)$ , the denominator $1 + e \cos ϕ$ is positive for $0 \leq ϕ < ϕ_{*}$ and tends to $0$ as $ϕ ↑ ϕ_{*}$ , so this conic locus is unbounded as well. These statements concern the full conic loci; no assertion is made that the actual time domain $I$ covers every point of either locus, or about asymptotic directions or scattering.

Remarks

Which items use the elliptic hypotheses. The completeness, closure and periodicity statements of clause 3(a) and the bounds of clause 3(b) hold only for $E < 0$ ; the conic formula of clause 1 and the barrier of clause 2 hold for every nonzero angular momentum and every energy reached by the trajectory.
The force centre and barycentre. In the reduced coordinate $r$ the focus is $r = 0$ , the force centre in relative-coordinate space. Clause 3(c) separately shows that each scaled coordinate $x_{i} - X$ has its focus at the centre of mass $X$ .
What is deliberately absent. No claim is made that the parabolic and hyperbolic branches have straight asymptotes, a deflection angle or a scattering cross-section on this page; those belong to the scattering development. No observation, measurement or empirical comparison appears here.

Physics theoremStatement: AI-adaptedProof: AI-adaptedOpen item page →

Kepler's third law for the gravitational two-label model: the relative period is $T = 2 π μ a b / J = 2 π \sqrt{μ / κ} a^{3 / 2}$ , so $T^{2} = 4 π^{2} a^{3} / (G_{N} M)$ under the mass identification, and both labels are periodic relative to the centre of mass

Physical scope

The period of the negative-energy gravitational relative orbit in the adopted two-label model: the relative motion is periodic, its period is $2 π μ a b / J = 2 π \sqrt{μ / κ} a^{3 / 2}$ , the form $T^{2} = 4 π^{2} a^{3} / (G_{N} M)$ holds when the gravitational couplings are identified with the inertial masses, each label is periodic about the centre of mass while its absolute motion advances by $V T$ per period, and the area swept per period is $π a b$ . The result is a conditional deduction from the adopted model and its declared empirical identification; it reports no observation and no measured value.

Statement

Assume the Axiom of Countable Choice (The Axiom of Countable Choice ( ${A C}_{ω}$ )). Adopt the two-label gravitational model of Adopted two-label gravitational model: the inverse-square pair potential with coupling parameters, the empirical mass identification, and the measured value and uncertainty of the Newtonian constant clause (i): inertial masses $m_{1}, m_{2} > 0$ , $M = m_{1} + m_{2}$ , reduced mass $μ = m_{1} m_{2} / M$ , gravitational couplings $m_{1}^{g}, m_{2}^{g} > 0$ , $κ = G_{N} m_{1}^{g} m_{2}^{g}$ , and the potential $u (ρ) = - κ / ρ$ on $(0, \infty)$ . Let $x : I \to Q$ be a motion on a time interval $I$ with relative coordinate $r$ , angular momentum $j = J e_{3}$ , $J > 0$ , and energy $E < 0$ , and let $e \in [0, 1)$ , $θ_{0} \in R$ , $p = \frac{J^{2}}{μ κ}, a = \frac{p}{1 - e^{2}} = - \frac{κ}{2 E}, b = a \sqrt{1 - e^{2}}$ be its conic data (The inverse-square gravitational relative trajectory has the Kepler polar form: $ρ = p / (1 + e \cos (θ - θ_{0}))$ , $E = (μ κ^{2} / 2 J^{2}) (e^{2} - 1)$ , elliptic completeness and barycentric scaled ellipses clauses 1 and 3). Then:

(Period of the relative motion.) The relative motion extends uniquely to a central-force trajectory on all of $R$ (as in The inverse-square gravitational relative trajectory has the Kepler polar form: $ρ = p / (1 + e \cos (θ - θ_{0}))$ , $E = (μ κ^{2} / 2 J^{2}) (e^{2} - 1)$ , elliptic completeness and barycentric scaled ellipses clause 3); along that extension (with $θ$ its polar angular coordinate, as supplied by the cited orbit theory), if $t_{0} \in R$ and $T > 0$ is determined by $θ (t_{0} + T) = θ (t_{0}) + 2 π$ , then $r (t + T) = r (t)$ and $r^{'} (t + T) = r^{'} (t)$ for all $t \in R$ , and $T = \frac{2 π μ a b}{J} = 2 π \sqrt{\frac{μ}{κ}} a^{3 / 2}, T^{2} = \frac{4 π^{2} μ a^{3}}{κ} .$
(Gravitational parameters.) With $μ = m_{1} m_{2} / M$ and $κ = G_{N} m_{1}^{g} m_{2}^{g}$ , $T^{2} = \frac{4 π^{2} a^{3} m_{1} m_{2}}{M G_{N} m_{1}^{g} m_{2}^{g}};$ if in addition the model adopts the identification $m_{i}^{g} = m_{i}$ of Adopted two-label gravitational model: the inverse-square pair potential with coupling parameters, the empirical mass identification, and the measured value and uncertainty of the Newtonian constant clause (ii), this reduces to $T^{2} = \frac{4 π^{2} a^{3}}{G_{N} M} .$
(Both labels.) On the global coordinate extension supplied by The inverse-square gravitational relative trajectory has the Kepler polar form: $ρ = p / (1 + e \cos (θ - θ_{0}))$ , $E = (μ κ^{2} / 2 J^{2}) (e^{2} - 1)$ , elliptic completeness and barycentric scaled ellipses clause 3(a) (write it again as $x$ , with centre of mass $X$ ), for each $i = 1, 2$ the position relative to the centre of mass is periodic with period $T$ , $x_{i} (t + T) - X (t + T) = x_{i} (t) - X (t)$ , while the absolute motions satisfy $x_{i} (t + T) - x_{i} (t) = V T (t \in R),$ with the constant centre-of-mass velocity $V = X^{'}$ of 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 clause 2; the absolute motions are periodic exactly when $V = 0$ . The restriction of this extension to the model time window $I_{0}$ is a model motion; statements about $x_{i} (t + T)$ within that model require $t, t + T \in I_{0}$ .
(Area per period.) The area swept by the relative radius vector in one period is $π a b$ , and it equals $J T / (2 μ)$ ; equivalently $T = 2 π μ a b / J$ as in clause 1.

Facts & Assumptions

Given: The gravitational two-label model with the constants $μ, κ, J > 0$ , the motion with $E < 0$ and its conic data $e, θ_{0}, p, a, b$ .

[A1]

The Axiom of Countable Choice is assumed (The Axiom of Countable Choice ( ${A C}_{ω}$ )).

[L1]

Reduction (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 clauses 2-4, Central-force trajectories: the central-force equation, angular momentum, energy and the effective potential, Adopted central-force model class: static central pair potentials, the reduced one-label problem, the fixed-centre idealization, reading conventions and exclusions clauses (i)-(ii)): the relative coordinate $r$ is a central-force trajectory for $(μ, u)$ with constant $j = J e_{3}$ and constant energy $E < 0$ ; the centre of mass is affine with constant $V = X^{'} = M^{- 1} P$ ; and $x_{i} - X = \pm \frac{m_{j}}{M} r$ for the two labels.

[L2]

Orbit theory (The inverse-square gravitational relative trajectory has the Kepler polar form: $ρ = p / (1 + e \cos (θ - θ_{0}))$ , $E = (μ κ^{2} / 2 J^{2}) (e^{2} - 1)$ , elliptic completeness and barycentric scaled ellipses clauses 1 and 3, Completeness and periodicity of the negative-energy Kepler orbit, with $T = 2 π μ a b / J$ and $T^{2} = 4 π^{2} μ a^{3} / κ$ , The subcritical Kepler conic is an ellipse with semiaxes $a = p / (1 - e^{2})$ , $b = a \sqrt{1 - e^{2}}$ , focus at the origin and area $π a b$ ; its area is also $\frac{1}{2} \int_{0}^{2 π} ϱ^{2}$ ): for the negative-energy inverse-square trajectory of [L1], the conic data $e \in [0, 1)$ , $p$ , $a = p / (1 - e^{2}) = - κ / (2 E)$ , $b = a \sqrt{1 - e^{2}}$ apply; the relative trajectory and the reconstructed coordinate curves extend uniquely to all of $R$ , with the coordinate curves restricted to $I_{0}$ defining the extended model motion, by clause 3(a) of the cited physical theorem, and the relative trajectory is periodic with $T$ determined by $θ (t_{0} + T) = θ (t_{0}) + 2 π$ and satisfying $T = 2 π μ a b / J = 2 π \sqrt{μ / κ} a^{3 / 2}$ , $T^{2} = 4 π^{2} μ a^{3} / κ$ ; and the ellipse of the conic has area $π a b = \frac{1}{2} \int_{0}^{2 π} ϱ (φ)^{2} d φ$ , where $ϱ (φ) = p / (1 + e \cos (φ - θ_{0}))$ .

[L3]

Area law (Planarity of the reduced orbit and the area law: the areal-velocity vector is the constant $j / (2 μ)$ , the swept-area rate is $J / (2 μ)$ , and both labels sweep equal areas in equal times about the centre of mass clauses 2-3): the swept-area function of the reduced orbit has constant derivative $J / (2 μ)$ , so the area swept over an interval of length $T$ is $J T / (2 μ)$ .

[L4]

Parameters (Adopted two-label gravitational model: the inverse-square pair potential with coupling parameters, the empirical mass identification, and the measured value and uncertainty of the Newtonian constant clauses (i)-(ii)): $κ = G_{N} m_{1}^{g} m_{2}^{g}$ and $μ = m_{1} m_{2} / M$ ; the identification $m_{i}^{g} = m_{i}$ is an adopted empirical assumption of the model's quantitative applications; $M = m_{1} + m_{2} > 0$ and $G_{N} > 0$ .

Proof

technique · direct

1.1givenA1L1L2algebra

Clause 1: by [L1] the relative trajectory meets the hypotheses of [L2] with $E < 0$ , and [L2] supplies the unique extension to $R$ , the periodicity with $T$ determined by one full revolution of $θ$ , and the two displayed period formulas; substituting $a = p / (1 - e^{2})$ and $p = J^{2} / (μ κ)$ into $T = 2 π μ a b / J$ and $T = 2 π \sqrt{μ / κ} a^{3 / 2}$ is exactly [L2].

2.1step 1.1L4algebra

Clause 2: substituting $μ = m_{1} m_{2} / M$ and $κ = G_{N} m_{1}^{g} m_{2}^{g}$ from [L4] into $T^{2} = 4 π^{2} μ a^{3} / κ$ gives $T^{2} = 4 π^{2} a^{3} m_{1} m_{2} / (M G_{N} m_{1}^{g} m_{2}^{g})$ ; if additionally $m_{i}^{g} = m_{i}$ , the numerator and denominator simplify by $m_{1} m_{2}$ to $T^{2} = 4 π^{2} a^{3} / (G_{N} M)$ .

2.2step 1.1L1L2algebra

Clause 3: use the global coordinate extension of [L2], which restricts to a model motion on $I_{0}$ . By [L1], $x_{i} - X = \pm \frac{m_{j}}{M} r$ , and $r$ , hence $x_{i} - X$ , is $T$ -periodic by step 1.1; also $X (t + T) - X (t) = V T$ because $X$ is affine with constant derivative $V$ , so $x_{i} (t + T) - x_{i} (t) = V T + (x_{i} (t + T) - X (t + T)) - (x_{i} (t) - X (t)) = V T$ . If $V = 0$ , this proves that both absolute motions are $T$ -periodic. Conversely, suppose either absolute motion $x_{i}$ has any positive period $S$ . For every positive integer $n$ , periodicity and the affine motion of $X$ give $0 = x_{i} (t + n S) - x_{i} (t) = n V S \pm \frac{m_{j}}{M} (r (t + n S) - r (t))$ for all $t \in R$ . By the conic bound in [L2], $∥ r (t) ∥_{2} \leq p / (1 - e)$ for all $t$ , so $n S ∥ V ∥_{2} \leq 2 (m_{j} / M) p / (1 - e)$ for every $n$ . Since the right side is fixed and $S > 0$ , this forces $V = 0$ . Thus either label has an absolute period only if $V = 0$ , and when $V = 0$ both are periodic.

3.1step 1.1L2L3algebra∎

Clause 4: by [L2] the relative motion extends to $R$ ; applying the relative areal-rate identity of [L3] to the global central-force trajectory, its swept-area function has derivative $J / (2 μ)$ , so the area swept over one period is $J T / (2 μ)$ . Since $θ : [t_{0}, t_{0} + T] \to [θ (t_{0}), θ (t_{0}) + 2 π]$ is a strictly increasing $C^{1}$ bijection with $θ^{'} = ρ^{- 2} μ^{- 1} J$ on the extension, the substitution rule gives $A (t_{0} + T) - A (t_{0}) = \frac{1}{2} \int_{t_{0}}^{t_{0} + T} ρ^{2} θ^{'} = \frac{1}{2} \int_{θ (t_{0})}^{θ (t_{0}) + 2 π} ϱ (φ)^{2} d φ = \frac{1}{2} \int_{0}^{2 π} ϱ (φ)^{2} d φ = π a b$ by the $2 π$ -periodicity of $ϱ$ and [L2]; combining with $A (t_{0} + T) - A (t_{0}) = J T / (2 μ)$ gives $T = 2 π μ a b / J$ .

Remarks

Which hypotheses carry the numerical form. Clauses 1, 3 and 4 use only the central model and the gravitational parameters symbolically; the simplification of clause 2 to $4 π^{2} a^{3} / (G_{N} M)$ uses the empirical identification of Adopted two-label gravitational model: the inverse-square pair potential with coupling parameters, the empirical mass identification, and the measured value and uncertainty of the Newtonian constant clause (ii), which is not a theorem of the framework.
No measured value. $a$ , $T$ , the masses and $G_{N}$ are model data or symbols; no observation, calibration or uncertainty is used here, and no numerical prediction is made.
Equal periods for both labels. Clause 3 states that both labels share the relative period about the centre of mass while the centre of mass may drift; the absolute motions are periodic only in the zero-total-momentum case.

RemarkRemark: AI-adaptedProof: Not applicableOpen item page →

Scope of the two-body and central-force page: collision and radial exclusions, point-particle and finite-size limits, the empirical parameters and their uncertainty, and the conditional status of the orbit statements

Physical scope

A scope and status audit of the two-body, central-force and gravitation page: which items are adopted assumptions, which are conditional deductions, which exclusions (collision, radial motion, finite size, external bodies, relativistic regime) the statements carry, how the empirical parameters $G_{N}$ and the mass identification qualify numerical conclusions, and which claims are deliberately not made. Bookkeeping and scope only; it asserts no law, no measurement and no conclusion beyond the items it cites.

Status audit of this page

This remark records the scope of the page's items. It is bookkeeping: every mathematical statement it repeats is proved in the item cited, and it adds no law, no observation and no numerical value of its own.

Adopted assumptions. The force branch is Adopted force model: force maps, the equation of motion, additive composition of force contributions and frame covariance with its standing regularity and equation of motion; the interaction restrictions (external-plus-pair presentation, antisymmetry, centrality, the potential branch and the gravitational branch) are Restricted interaction models: pair decompositions, antisymmetry and centrality options, potential forces and the gravitational branch; the page's central-force class, the reduced problem, the fixed-centre idealization and the reading conventions are Adopted central-force model class: static central pair potentials, the reduced one-label problem, the fixed-centre idealization, reading conventions and exclusions; and the gravitational model with its mass identification and the measured value of $G_{N}$ is Adopted two-label gravitational model: the inverse-square pair potential with coupling parameters, the empirical mass identification, and the measured value and uncertainty of the Newtonian constant. None of these is proved here.

Conditional deductions. The relative reduction, the uniform centre-of-mass motion and the reconstruction bijection are 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 planarity and area law are Planarity of the reduced orbit and the area law: the areal-velocity vector is the constant $j / (2 μ)$ , the swept-area rate is $J / (2 μ)$ , and both labels sweep equal areas in equal times about the centre of mass; the effective one-dimensional problem and its converse are Reduction of the central problem to the effective one-dimensional potential: $μ ρ^{''} = - (U_{J}^{e f f})^{'} (ρ)$ , the energy identity, and the converse reconstruction of the planar trajectory from the radial data; the circular-orbit criterion and the linearised radial frequency are Circular relative orbits of the central model: the critical-point criterion $u^{'} (r_{0}) = J^{2} / (μ r_{0}^{3})$ , uniform rotation of both labels about the centre of mass, and the linearised radial frequency; the conic classification, the barrier and the elliptic geometry are The inverse-square gravitational relative trajectory has the Kepler polar form: $ρ = p / (1 + e \cos (θ - θ_{0}))$ , $E = (μ κ^{2} / 2 J^{2}) (e^{2} - 1)$ , elliptic completeness and barycentric scaled ellipses; and the period and third law are Kepler's third law for the gravitational two-label model: the relative period is $T = 2 π μ a b / J = 2 π \sqrt{μ / κ} a^{3 / 2}$ , so $T^{2} = 4 π^{2} a^{3} / (G_{N} M)$ under the mass identification, and both labels are periodic relative to the centre of mass. Each is a conditional consequence of the adopted model and the cited mathematics. The conic and period items assume the Axiom of Countable Choice (The Axiom of Countable Choice ( ${A C}_{ω}$ )); every scope statement below that invokes their conclusions carries that assumption.

Exclusions and limits

Collision and the domain. The configuration domain excludes $x_{1} = x_{2}$ , i.e. $r = 0$ (Configuration domain of $N$ labeled particles: open admissible sets, regular holonomic constraints and the mass operator, Adopted central-force model class: static central pair potentials, the reduced one-label problem, the fixed-centre idealization, reading conventions and exclusions clause (v)); the equations of motion and every orbit statement above are undefined at the collision. In the gravitational model with $j \neq 0$ the barrier of The inverse-square gravitational relative trajectory has the Kepler polar form: $ρ = p / (1 + e \cos (θ - θ_{0}))$ , $E = (μ κ^{2} / 2 J^{2}) (e^{2} - 1)$ , elliptic completeness and barycentric scaled ellipses clause 2 keeps $∣ r ∣ \geq ρ_{\min} (E) > 0$ throughout the motion; no such bound is available for the radial case, so the conic, area and period statements must not be extended to it.
Radial motion. For $j = 0$ the nonzero-angular-momentum polar parametrization and its distinguished plane orthogonal to $j$ are unavailable; a constant polar angle can still be chosen in any plane containing the fixed radial direction. Planarity of the reduced orbit and the area law: the areal-velocity vector is the constant $j / (2 μ)$ , the swept-area rate is $J / (2 μ)$ , and both labels sweep equal areas in equal times about the centre of mass clauses 2-4, The inverse-square gravitational relative trajectory has the Kepler polar form: $ρ = p / (1 + e \cos (θ - θ_{0}))$ , $E = (μ κ^{2} / 2 J^{2}) (e^{2} - 1)$ , elliptic completeness and barycentric scaled ellipses and Kepler's third law for the gravitational two-label model: the relative period is $T = 2 π μ a b / J = 2 π \sqrt{μ / κ} a^{3 / 2}$ , so $T^{2} = 4 π^{2} a^{3} / (G_{N} M)$ under the mass identification, and both labels are periodic relative to the centre of mass exclude this case explicitly. Clause 1 of Reduction of the central problem to the effective one-dimensional potential: $μ ρ^{''} = - (U_{J}^{e f f})^{'} (ρ)$ , the energy identity, and the converse reconstruction of the planar trajectory from the radial data also assumes $J = ∣ j ∣ > 0$ , so it does not supply the radial equation. Instead, angular-momentum conservation in Planarity of the reduced orbit and the area law: the areal-velocity vector is the constant $j / (2 μ)$ , the swept-area rate is $J / (2 μ)$ , and both labels sweep equal areas in equal times about the centre of mass clause 1 and the relative equation of 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 show that, when $j = 0$ , $r^{'} = λ r$ with $λ = ⟨ r, r^{'} ⟩ / ∣ r ∣^{2}$ . Setting $ρ = ∣ r ∣ > 0$ gives $ρ^{'} = λ ρ$ and $n^{'} = r^{'} / ρ - r ρ^{'} / ρ^{2} = 0$ for $n = r / ρ$ ; hence the unit direction is constant, say $n_{0}$ , and $r = ρ n_{0}$ with $ρ \in J_{0}$ . Substitution into that relative equation yields the scalar equation $μ ρ^{''} = - u^{'} (ρ)$ on $J_{0}$ . No centrifugal term, equal-area, conic or period claim is made for radial trajectories.
Point particles and finite size. The model retains only positions and fixed masses of the labels (Particle labels, worldlines, the frame-time parameter condition and fixed inertial masses clause (vi)); size, shape, rotation, deformation, internal structure and mass loss are omitted. A real body whose radius, multipole moments, tides or spin matter for the question asked is not described by these items, and their applicability must be justified separately wherever they are used.
External bodies and non-gravitational forces. The model has no external forces by construction (Adopted central-force model class: static central pair potentials, the reduced one-label problem, the fixed-centre idealization, reading conventions and exclusions clause (i)), so third bodies, drag, radiation pressure or any other agent are outside it; a two-body computation with such influences is a different model, not a corollary of this page.
Regime. The meanings of weak-field and non-relativistic use are those defined in Adopted two-label gravitational model: the inverse-square pair potential with coupling parameters, the empirical mass identification, and the measured value and uncertainty of the Newtonian constant clause (iv): for a trajectory segment and chosen tolerance $η \in (0, 1)$ , in the fixed comparison inertial frame of that clause, the maximum source-potential ratio $\max_{i} ∣ Φ_{i} ∣ / c^{2}$ and the maximum squared label-speed ratio $\max_{i} ∣ x_{i}^{'} ∣^{2} / c^{2}$ are each at most $η$ throughout the segment. Relativistic corrections (for instance the perihelion precession of the inner planets) require the general-relativistic framework and are neither derived nor bounded here; the inverse-square law itself is an adopted law of the model.

Empirical status

The constant. Formula-level statements use the symbol $G_{N}$ and are conditional on Adopted two-label gravitational model: the inverse-square pair potential with coupling parameters, the empirical mass identification, and the measured value and uncertainty of the Newtonian constant clause (i). Any numerical application that assigns the CODATA value of clause (iii) — relative standard uncertainty $2.2 \times 10^{- 5}$ , with the mutual inconsistency of the 16 input measurements recorded there — inherits that uncertainty and the conditions of every other input actually used. The framework does not derive the value of $G_{N}$ and does not claim it is exact.
The mass identification. The simplification of the third law to $T^{2} = 4 π^{2} a^{3} / (G_{N} M)$ in Kepler's third law for the gravitational two-label model: the relative period is $T = 2 π μ a b / J = 2 π \sqrt{μ / κ} a^{3 / 2}$ , so $T^{2} = 4 π^{2} a^{3} / (G_{N} M)$ under the mass identification, and both labels are periodic relative to the centre of mass clause 2 uses the empirical identification $m_{i}^{g} = m_{i}$ of Adopted two-label gravitational model: the inverse-square pair potential with coupling parameters, the empirical mass identification, and the measured value and uncertainty of the Newtonian constant clause (ii), which is an adopted assumption, not a consequence of the equation of motion. Statements that keep the couplings $m_{i}^{g}$ distinct remain valid without it.
No local experiment. No item of this page reports an experiment or apparatus record. Adopted two-label gravitational model: the inverse-square pair potential with coupling parameters, the empirical mass identification, and the measured value and uncertainty of the Newtonian constant clause (iii) does report the CODATA 2022 recommended estimate and standard uncertainty for $G_{N}$ , the source's 16 measured input values, and the 3.9 uncertainty-expansion factor. Those are source-reported aggregate evidence, not a measurement performed or newly reported by this page; the individual experimental apparatus records are not reproduced here. The numerical checks on the companion examples page use declared model data. Comparing any prediction of this page with observations would test the combined model, parameter and apparatus assumptions; agreement would be evidence, not a proof of the model, and a discrepancy would identify no particular postulate by itself.

Deliberately not claimed

No global existence of arbitrary central-force trajectories, no nonlinear stability of circular orbits, no closed-form solution of the general non-Coulomb central problem, no asymptotes or scattering cross-sections for the unbounded conic branches, no relativistic or finite-size corrections, and no claim that any physical system is exactly a model of the adopted class.

5 · Examples, experiments and counterexamples

None yet.

Two-body, central forces and gravitation

1 · Prerequisites

2 · Summary

3 · Logical flowchart

4 · Definitions, postulates, theorems and proofs

The radial composite x↦u(∣x∣)x↦u(∣x∣) on an annulus: CkCk regularity and ∇(u∘∣⋅∣)(x)=u′(∣x∣) x/∣x∣∇(u∘∣⋅∣)(x)=u′(∣x∣)x/∣x∣

Statement

Facts & Assumptions

Proof

Remarks

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

Statement

Facts & Assumptions

Proof

Remarks

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

Postulate

Status and scope

Planarity of the reduced orbit and the area law: the areal-velocity vector is the constant j/(2μ)j/(2μ), the swept-area rate is J/(2μ)J/(2μ), and both labels sweep equal areas in equal times about the centre of mass

Statement

Facts & Assumptions

Proof

Remarks

Reduction of the central problem to the effective one-dimensional potential: μρ′′=−(UJeff)′(ρ)μρ′′=−(UJeff​)′(ρ), the energy identity, and the converse reconstruction of the planar trajectory from the radial data

Statement

Facts & Assumptions

Proof

Remarks

Circular relative orbits of the central model: the critical-point criterion u′(r0)=J2/(μr03)u′(r0​)=J2/(μr03​), uniform rotation of both labels about the centre of mass, and the linearised radial frequency

Statement

Facts & Assumptions

Proof

Remarks

Adopted two-label gravitational model: the inverse-square pair potential with coupling parameters, the empirical mass identification, and the measured value and uncertainty of the Newtonian constant

Postulate

Status and scope

The inverse-square gravitational relative trajectory has the Kepler polar form: ρ=p/(1+ecos⁡(θ−θ0))ρ=p/(1+ecos(θ−θ0​)), E=(μκ2/2J2)(e2−1)E=(μκ2/2J2)(e2−1), elliptic completeness and barycentric scaled ellipses

Statement

Facts & Assumptions

Proof

Remarks

Kepler's third law for the gravitational two-label model: the relative period is T=2πμab/J=2πμ/κ a3/2T=2πμab/J=2πμ/κ​a3/2, so T2=4π2a3/(GNM)T2=4π2a3/(GN​M) under the mass identification, and both labels are periodic relative to the centre of mass

Statement

Facts & Assumptions

Proof

Remarks

Scope of the two-body and central-force page: collision and radial exclusions, point-particle and finite-size limits, the empirical parameters and their uncertainty, and the conditional status of the orbit statements

Status audit of this page

Exclusions and limits

Empirical status

Deliberately not claimed

5 · Examples, experiments and counterexamples

Sources

The radial composite $x \mapsto u (∣ x ∣)$ on an annulus: $C^{k}$ regularity and $\nabla (u \circ ∣ \cdot ∣) (x) = u^{'} (∣ x ∣) x / ∣ x ∣$

Planarity of the reduced orbit and the area law: the areal-velocity vector is the constant $j / (2 μ)$ , the swept-area rate is $J / (2 μ)$ , and both labels sweep equal areas in equal times about the centre of mass

Reduction of the central problem to the effective one-dimensional potential: $μ ρ^{''} = - (U_{J}^{e f f})^{'} (ρ)$ , the energy identity, and the converse reconstruction of the planar trajectory from the radial data

Circular relative orbits of the central model: the critical-point criterion $u^{'} (r_{0}) = J^{2} / (μ r_{0}^{3})$ , uniform rotation of both labels about the centre of mass, and the linearised radial frequency

The inverse-square gravitational relative trajectory has the Kepler polar form: $ρ = p / (1 + e \cos (θ - θ_{0}))$ , $E = (μ κ^{2} / 2 J^{2}) (e^{2} - 1)$ , elliptic completeness and barycentric scaled ellipses

Kepler's third law for the gravitational two-label model: the relative period is $T = 2 π μ a b / J = 2 π \sqrt{μ / κ} a^{3 / 2}$ , so $T^{2} = 4 π^{2} a^{3} / (G_{N} M)$ under the mass identification, and both labels are periodic relative to the centre of mass