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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.

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

Statement

Facts & Assumptions

Proof

Remarks

Depends on

Dependency tree · 2 levels

Sources

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

Depends on

Dependency tree · 2 levels

Sources

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