Topology and bifurcations of Kirchhoff solution submanifolds

Characterize the topology of the three-dimensional submanifolds defined by fixed energy, squared linear momentum, and angular invariant for the Kirchhoff equations, and determine the associated bifurcations as the conserved quantities vary.

Background

The Kirchhoff equations possess three conserved quantities: kinetic energy, the squared magnitude of linear momentum, and the angular invariant given by the scalar product of angular and linear momentum. Their common level sets form three-dimensional submanifolds in the six-dimensional momentum space on which the dynamics evolve. For the Aref–Jones ellipsoid, the authors note that the topology of these intersections and the changes induced by varying the conserved quantities have not been established. This problem concerns the global geometric organization of the solution space rather than the stability of individual landmark solutions.

References

The topology of these submanifolds, and the associated bifurcations as the quantities are varied, are to our knowledge unknown.

— Some explorations of Kirchhoff dynamics  (2609.00509 - Oguns et al., 1 Sep 2026) in Section 2.1, “The Aref-Jones ellipsoid and its solution manifold”

The seeming simplicity of Zone 4, which we did not explore as all tumbling states were stable, might actually belie something more interesting, as it is not obvious how these stable orbits meet up to fill the space.

— Some explorations of Kirchhoff dynamics  (2609.00509 - Oguns et al., 1 Sep 2026) in Section 7, Discussion

It is possible that other such bifurcations exist in other zones, but are not easy to see, and such questions would benefit from a more systematic mathematical approach to the topology of the intersections of conserved quantity level sets.

— Some explorations of Kirchhoff dynamics  (2609.00509 - Oguns et al., 1 Sep 2026) in Section 7, Discussion