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A zero-angular-momentum Kepler trajectory reaches the force centre in finite time, refuting unconditional global existence

Statement refuted

False statement. Every solution of the attractive inverse-square equation $μ r^{''} = - κ r / ∣ r ∣^{3}$ ( $μ, κ > 0$ ) on its maximal interval of definition is defined for all real times and satisfies $∣ r (t) ∣ \geq c$ for some $c > 0$ .

The witness has zero angular momentum and reaches the excluded force centre, so the non-radial positive results do not apply. Its inverse-square potential is singular and is not finite-range; this example does not establish the necessity of the simple-turning-point hypothesis in Finite-range scattering: the simple outer turning point, the two global branches and the free tails.

Facts & Assumptions

Given: Parameters $μ > 0$ , $κ > 0$ , the constant $A : = (9 κ / (2 μ))^{1 / 3} > 0$ , and the curve $r : (- \infty, 1) \to R^{3}$ , $r (t) : = A (1 - t)^{2 / 3} e_{1}$ , where $e_{1}$ is a unit vector.

[L1]

The central-force equation in the radial case: for a curve $r = ρ e_{1}$ with $ρ > 0$ of class $C^{2}$ , the equation $μ r^{''} = - κ r / ∣ r ∣^{3}$ is equivalent to $μ ρ^{''} = - κ / ρ^{2}$ , and the angular momentum $μ r \times r^{'}$ vanishes identically (Central-force trajectories: the central-force equation, angular momentum, energy and the effective potential, $C^{k}$ Euclidean maps and diffeomorphisms).

[L2]

Power derivatives: for $t < 1$ , $\frac{d}{d t} (1 - t)^{2 / 3} = - \frac{2}{3} (1 - t)^{- 1 / 3}$ and $\frac{d}{d t} (- \frac{2}{3} (1 - t)^{- 1 / 3}) = - \frac{2}{9} (1 - t)^{- 4 / 3}$ , by the real-power derivative and the chain rule (Continuity and derivatives of positive-base real powers, 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 $g (c) \neq 0$ , The derivative $f^{'} (c) = \lim_{x \to c} \frac{f (x) - f (c)}{x - c}$ of $f : A \to R$ at a point $c \in A$ that is a limit point of $A$ , and differentiability on a set, Limit point, isolated point, adherent point, derived set, and dense subset of $R$ , Intervals of $R$ : the nine order-convex forms, nondegeneracy, and length).

[L3]

For $t < 1$ , $A^{3} (1 - t)^{2} = \frac{9 κ}{2 μ} (1 - t)^{2}$ and, equivalently, $\frac{2 μ A}{9} = \frac{κ}{A^{2}}$ : this is the defining relation of $A$ , and it is an equality of positive real numbers (multiply $A^{3} = 9 κ / (2 μ)$ by $2 μ / (9 A^{2})$ ).

Counterexample

technique · direct

1.1givenL1L2L3algebra

The curve solves the equation: by [L2], $r^{''} (t) = - \frac{2 A}{9} (1 - t)^{- 4 / 3} e_{1}$ , so $μ r^{''} = - \frac{2 μ A}{9} (1 - t)^{- 4 / 3} e_{1}$ , while $- κ r / ∣ r ∣^{3} = - κ ρ^{- 2} e_{1} = - \frac{κ}{A^{2}} (1 - t)^{- 4 / 3} e_{1}$ because $ρ (t) = A (1 - t)^{2 / 3}$ . By [L3] the two coefficients agree, so $μ r^{''} = - κ r / ∣ r ∣^{3}$ for every $t < 1$ ; and the angular momentum vanishes identically.

2.1givenstep 1.1L1algebra

Failure of global existence: $ρ (t) = A (1 - t)^{2 / 3} \to 0$ as $t ↑ 1$ , and $ρ$ is strictly positive and $C^{2}$ on $(- \infty, 1)$ . There is no solution of $μ r^{''} = - κ r / ∣ r ∣^{3}$ on an interval containing $1$ whose restriction to $(- \infty, 1)$ equals $r$ : such a solution would be continuous at $1$ with $r (1) = \lim_{t ↑ 1} r (t) = 0$ , and the equation is undefined at the origin (the right-hand side contains $r / ∣ r ∣^{3}$ ). Hence the maximal interval of $r$ is $(- \infty, 1)$ , not $R$ .

3.1step 1.1step 2.1algebra∎

Failure of the lower bound: $\inf_{t < 1} ρ (t) = 0$ , so there is no $c > 0$ with $∣ r (t) ∣ \geq c$ on the domain. Together with step 2.1, this refutes the stated universal claim.

Depends on

Dependency tree · 2 levels

Sources

David Tong, Classical Dynamics (University of Cambridge), §2.5.4, printed pp. 32-33: the angular-momentum barrier and the separation of the radial and angular problems