Self-similarity in the Kepler-Heisenberg problem

Corey Shanbrom; Victor Dods

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Self-similarity in the Kepler-Heisenberg problem

Victor Dods, Corey Shanbrom

Dynamical Systems · Submitted 28 Dec 2019

abstract compiled · 2026-09-23 · no language-model review

The authors' main claim Here we demonstrate that, under mild conditions, zero-energy orbits are self-similar.

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Compiled by classifier-1.13.0 via TypeSafe API, question bank abstract-p1-20260923, 2026-09-23: 1 request, 5,539 input tokens.

What the abstract itself states, as probabilities: claims first 0.04 · states limitation 0.48 · human participants 0.01 · controlled comparison 0.01 · randomized assignment 0.01 · states quantitative result 0.27 · compares with prior results 0.20 · mentions code or data availability 0.02.

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arXiv listing abstract v2. Results and the main claim are underlined; show every role.

The Kepler-Heisenberg problem is that of determining the motion of a planet around a sun in the Heisenberg group, thought of as a three-dimensional sub-Riemannian manifold. The sub-Riemannian Hamiltonian provides the kinetic energy, and the gravitational potential is given by the fundamental solution to the sub-Laplacian. The dynamics are at least partially integrable, possessing two first integrals as well as a dilational momentum which is conserved by orbits with zero energy. The system is known to admit closed orbits of any rational rotation number, which all lie within the fundamental zero-energy integrable subsystem. Here we demonstrate that, under mild conditions, zero-energy orbits are self-similar. Consequently we find that these zero-energy orbits stratify into three families: future collision, past collision, and quasi-periodicity without collision. If a collision occurs, it occurs in finite time.

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What the classifier reports for each sentence. The number is the probability it gave the label.
#	sentence	role	the authors' own work	states the main conclusion	strength
s0	The Kepler-Heisenberg problem is that of determining the motion of a planet around a sun in the Heisenberg group, thought of as a three-dimensional sub-Riemannian manifold.	background 1.00	0.05	0.02	asserted 1.00
s1	The sub-Riemannian Hamiltonian provides the kinetic energy, and the gravitational potential is given by the fundamental solution to the sub-Laplacian.	background 0.87	0.09	0.02	asserted 1.00
s2	The dynamics are at least partially integrable, possessing two first integrals as well as a dilational momentum which is conserved by orbits with zero energy.	background 0.83	0.15	0.04	asserted 0.80
s3	The system is known to admit closed orbits of any rational rotation number, which all lie within the fundamental zero-energy integrable subsystem.	background 0.99	0.06	0.04	asserted 1.00
s4	Here we demonstrate that, under mild conditions, zero-energy orbits are self-similar.	result 0.68	0.95	0.81	asserted 1.00
s5	Consequently we find that these zero-energy orbits stratify into three families: future collision, past collision, and quasi-periodicity without collision.	result 0.90	0.94	0.26	asserted 1.00
s6	If a collision occurs, it occurs in finite time.	result 0.99	0.68	0.08	asserted 1.00

What kind of paper the classifier reports this is: theoretical result 1.00 · evidence from proof or derivation 1.00.

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Original abstract

The Kepler-Heisenberg problem is that of determining the motion of a planet around a sun in the Heisenberg group, thought of as a three-dimensional sub-Riemannian manifold. The sub-Riemannian Hamiltonian provides the kinetic energy, and the gravitational potential is given by the fundamental solution to the sub-Laplacian. The dynamics are at least partially integrable, possessing two first integrals as well as a dilational momentum which is conserved by orbits with zero energy. The system is known to admit closed orbits of any rational rotation number, which all lie within the fundamental zero-energy integrable subsystem. Here we demonstrate that, under mild conditions, zero-energy orbits are self-similar. Consequently we find that these zero-energy orbits stratify into three families: future collision, past collision, and quasi-periodicity without collision. If a collision occurs, it occurs in finite time.

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