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arXiv · 2601.06416

Symplectic mechanics of relativistic spinning compact bodies. III. quadratic-in-spin integrability in Type-D Einstein spacetimes: persistence and breakdown

Paul Ramond·Soichiro Isoyama·Adrien Druart
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Abstract

We investigate the integrability of spinning compact body dynamics at quadratic order in spin in four-dimensional Einstein spacetimes admitting a non-degenerate Killing--Yano tensor. Working within the Mathisson--Papapetrou--Tulczyjew--Dixon framework under the Tulczyjew--Dixon spin supplementary condition, we model the spin-induced quadrupole with a deformability parameter $\kappa$, where $\kappa=1$ corresponds to black holes. The dynamics is formulated as a Hamiltonian system on a 10-dimensional physical phase space obtained by Dirac--Bergmann reduction. For $\kappa=1$, we establish Liouville--Arnold integrability at quadratic order in spin by constructing five independent, Poisson-commuting first integrals, including a generalization of the Carter constant and the R\"udiger constant to quadratic-in-spin order in Einstein spacetimes beyond Kerr. For $\kappa \neq 1$, the R\"udiger and Carter constants are no longer conserved; integrability does not persist at this order. All our results are carried out in a covariant manner and numerically verified, and Kerr is recovered as a special case. These results show that integrability can extend beyond Kerr and beyond the linear-in-spin regime, while its breakdown for $\kappa \neq 1$ points to the spin-induced quadrupole as a decisive probe of compact body structure.

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Paul Ramond, Soichiro Isoyama, Adrien Druart. 2026-01-10. Symplectic mechanics of relativistic spinning compact bodies. III. quadratic-in-spin integrability in Type-D Einstein spacetimes: persistence and breakdown. https://arxiv.org/abs/2601.06416

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