Relative Periodic Orbits in the Gutzwiller-type Anisotropic Kepler Problem and $n$-body Problem

Abstract: We study the relative periodic orbits in the Gutzwiller-type anisotropic Kepler problem, which is a generalized model derived from the classical Gutzwiller anisotropic Kepler problem. By reducing the rotational symmetry of the $z$-axis, we obtain a reduced system with two degrees of freedom and its energy surface forms a compact and regular three-sphere in a certain parameter range. Combining the estimation of the Seifert rotation number of the planar Kepler orbit and the CHHL formula introduced in \cite{CHHL23}, we prove that this system admits infinitely many periodic orbits on every compact regular energy surface. This model can be applied to the $(1+2n)$-body problem to find infinitely many relative periodic orbits with hip-hop symmetry as well as relative periodic orbits in the $n$-pyramidal problem.

• The paper proves infinitely many periodic orbits on every compact regular energy surface when the effective anisotropy satisfies β=D/C−1≥0, using contact-volume estimates, rotation numbers, and the CHHL two-or-infinity theorem.
• The analysis shows that the planar Kepler orbit’s rotation number strictly exceeds the volume threshold except in the integrable spatial Kepler case, where equality occurs.
• The result guarantees infinitely many hip-hop relative periodic orbits for all central mass ratios and extends the n-pyramidal conclusion to all mass ratios for 2≤n≤472, while leaving the D<C regime open.

Overview and main result

This paper by Hu, Ou, Qiao, and Yang (2608.12901) studies the reduced Gutzwiller-type anisotropic Kepler problem, a two-degree-of-freedom Hamiltonian system of the form

$$H(p_r,p_z,r,z)=\tfrac{1}{2}(p_r^2+p_z^2)+\frac{\varpi^2}{2r^2}-\frac{A_0}{r}-\sum_{i=1}^{\mathfrak n}\frac{A_i}{\sqrt{r^2+B_iz^2}},$$

with $A_0\ge 0$ and $A_i,B_i>0$. The classical Gutzwiller anisotropic Kepler problem corresponds to $A_0=0$, $\mathfrak n=1$. The main theorem states that, under the condition $$\sum_{i=1}^{\mathfrak n}A_iB_i\ge \sum_{i=0}^{\mathfrak n}A_i$$ (equivalently $\beta=D/C-1\ge 0$ with $D=\sum A_iB_i$, $C=\sum A_i$), every compact regular energy surface admits infinitely many periodic orbits. The proof combines a contact-volume estimate, an estimate of the Seifert rotation number of the planar Kepler orbit via Maslov-type index theory, and the CHHL formula from the resolution of the two-or-infinity conjecture [CHHL23, CHHL26].

The strategy is a comparison argument: if the CHHL formula held with only two simple Reeb orbits, one would have $\mathrm{vol}(\mathfrak M,\lambda)=T_p^2/\hat\rho_p$ for the planar Kepler orbit $A_0\ge 0$0. The paper shows instead that $A_0\ge 0$1 strictly (with equality only in the integrable spatial Kepler case), forcing infinitely many Reeb orbits.

Topology of energy surfaces and the reduction framework

A key structural parameter is $A_0\ge 0$2, where $A_0\ge 0$3 is the energy level and $A_0\ge 0$4 the conserved angular momentum about the $A_0\ge 0$5-axis. The paper classifies the topology completely: when $A_0\ge 0$6, the energy surface $A_0\ge 0$7 is compact and homeomorphic to $A_0\ge 0$8; at $A_0\ge 0$9 it degenerates to a single point (the non-degenerate global minimum); below this value it is empty; and for $A_i,B_i>0$0 it is unbounded in the $A_i,B_i>0$1-direction. In the compact range, $A_i,B_i>0$2 is a contact-type hypersurface in $A_i,B_i>0$3, so the Hamiltonian flow coincides with a Reeb flow up to reparametrization.

The planar Kepler orbit $A_i,B_i>0$4 is explicit, with eccentricity $A_i,B_i>0$5, and serves as the distinguished Reeb orbit whose rotation number is compared against the contact volume.

Contact volume estimates

The volume analysis reduces the general system to a comparison anisotropic Kepler problem with a single effective anisotropy $A_i,B_i>0$6. Using strict convexity of $A_i,B_i>0$7 and Jensen's inequality, the authors show

$A_i,B_i>0$8

with equality if and only if $A_i,B_i>0$9 and all $A_0=0$0 coincide — i.e., precisely when the system is the reduced spatial Kepler problem. This volume bound is sharp in exactly the integrable case, which is what makes the subsequent strict rotation-number inequality decisive.

Rotation number via the Hill stability equation

The linearized flow along $A_0=0$1 decouples into two subsystems. The transverse one, after a time-dependent symplectic change of variables, becomes the Hill stability equation

$A_0=0$2

a special case of Ince's equation that also governs the stability of homographic solutions in the $A_0=0$3-body problem. Its Seifert rotation number $A_0=0$4 is analyzed through the $A_0=0$5-degenerate curves $A_0=0$6 of the associated self-adjoint Fredholm operator.

For $A_0=0$7, the paper extends a previously known inequality (valid for $A_0=0$8) to all $A_0=0$9 by proving, via a blow-up technique near collision singularities and Sturm comparison applied to the coefficient function $\mathfrak n=1$0, that

$\mathfrak n=1$1

where the explicit threshold $\mathfrak n=1$2 satisfies $\mathfrak n=1$3 on $\mathfrak n=1$4. Combining this with the volume estimate yields $\mathfrak n=1$5 whenever $\mathfrak n=1$6, and also in the degenerate case $\mathfrak n=1$7 with genuine anisotropy ($\mathfrak n=1$8 or unequal $\mathfrak n=1$9), where $$\sum_{i=1}^{\mathfrak n}A_iB_i\ge \sum_{i=0}^{\mathfrak n}A_i$$0 but the volume strictly exceeds $$\sum_{i=1}^{\mathfrak n}A_iB_i\ge \sum_{i=0}^{\mathfrak n}A_i$$1. By the corollary of the CHHL formula, each compact regular energy surface carries infinitely many periodic orbits. Equality holds only for the integrable Kepler case, consistent with the earlier results of Hu–Ou–Qiao covering $$\sum_{i=1}^{\mathfrak n}A_iB_i\ge \sum_{i=0}^{\mathfrak n}A_i$$2; together with a conformal symplectic rescaling handling $$\sum_{i=1}^{\mathfrak n}A_iB_i\ge \sum_{i=0}^{\mathfrak n}A_i$$3, infinitely many periodic orbits are now established for the classical anisotropic Kepler problem for all $$\sum_{i=1}^{\mathfrak n}A_iB_i\ge \sum_{i=0}^{\mathfrak n}A_i$$4.

Applications to the hip-hop $$\sum_{i=1}^{\mathfrak n}A_iB_i\ge \sum_{i=0}^{\mathfrak n}A_i$$5-body problem

The hip-hop symmetric invariant subsystem of the $$\sum_{i=1}^{\mathfrak n}A_iB_i\ge \sum_{i=0}^{\mathfrak n}A_i$$6-body problem — $$\sum_{i=1}^{\mathfrak n}A_iB_i\ge \sum_{i=0}^{\mathfrak n}A_i$$7 equal masses forming two regular $$\sum_{i=1}^{\mathfrak n}A_iB_i\ge \sum_{i=0}^{\mathfrak n}A_i$$8-gons reflected across planes perpendicular to the $$\sum_{i=1}^{\mathfrak n}A_iB_i\ge \sum_{i=0}^{\mathfrak n}A_i$$9-axis, plus a central mass of ratio $\beta=D/C-1\ge 0$0 fixed at the center of mass — reduces after fixing angular momentum to a Hamiltonian of the Gutzwiller type with $\beta=D/C-1\ge 0$1, $\beta=D/C-1\ge 0$2, $\beta=D/C-1\ge 0$3, $\beta=D/C-1\ge 0$4, and $\beta=D/C-1\ge 0$5, $\beta=D/C-1\ge 0$6. The planar Kepler orbit here is the elliptic relative equilibrium (ERE) of the regular $\beta=D/C-1\ge 0$7-gon central configuration.

Verifying the hypothesis $\beta=D/C-1\ge 0$8 amounts to showing $\beta=D/C-1\ge 0$9, where $D=\sum A_iB_i$0 and $D=\sum A_iB_i$1. The authors prove this for all $D=\sum A_iB_i$2 and $D=\sum A_iB_i$3 using the lower bound $D=\sum A_iB_i$4, which exceeds $D=\sum A_iB_i$5 already at $D=\sum A_iB_i$6 and diverges as $D=\sum A_iB_i$7. Consequently, every compact regular energy surface of the hip-hop system carries infinitely many relative periodic orbits with hip-hop symmetry under Newtonian gravity, for all central mass ratios. This strengthens Terracini–Venturelli's variational existence result for weak-force potentials: their construction did not control which energy surface the orbits lie on, whereas the present result gives orbits on every compact regular energy surface. The paper notes, however, that the total rotation angle about the $D=\sum A_iB_i$8-axis over one period cannot be determined for any given orbit — a limitation inherited from the method.

Applications to the $D=\sum A_iB_i$9-pyramidal problem

For the $C=\sum A_i$0-pyramidal problem ($C=\sum A_i$1 masses, $C=\sum A_i$2 identical masses in a regular $C=\sum A_i$3-gon symmetric about a fixed axis along which the first body moves), the reduced Hamiltonian fits the framework with $C=\sum A_i$4, $C=\sum A_i$5, $C=\sum A_i$6, $C=\sum A_i$7, where $C=\sum A_i$8 is the mass ratio. Here the hypothesis $C=\sum A_i$9 becomes $\mathrm{vol}(\mathfrak M,\lambda)=T_p^2/\hat\rho_p$0, i.e. $\mathrm{vol}(\mathfrak M,\lambda)=T_p^2/\hat\rho_p$1. Since $\mathrm{vol}(\mathfrak M,\lambda)=T_p^2/\hat\rho_p$2 grows like $\mathrm{vol}(\mathfrak M,\lambda)=T_p^2/\hat\rho_p$3, this holds for $\mathrm{vol}(\mathfrak M,\lambda)=T_p^2/\hat\rho_p$4 — a concrete numerical threshold. Within this range, every compact regular energy surface admits infinitely many periodic orbits for all mass ratios $\mathrm{vol}(\mathfrak M,\lambda)=T_p^2/\hat\rho_p$5, extending prior results that required either sufficiently large mass ratio or sufficiently small $\mathrm{vol}(\mathfrak M,\lambda)=T_p^2/\hat\rho_p$6. For $\mathrm{vol}(\mathfrak M,\lambda)=T_p^2/\hat\rho_p$7 the hypothesis fails and the question remains open.

Limitations and open questions

Several restrictions are intrinsic to the approach. The main theorem requires $\mathrm{vol}(\mathfrak M,\lambda)=T_p^2/\hat\rho_p$8; systems with $\mathrm{vol}(\mathfrak M,\lambda)=T_p^2/\hat\rho_p$9 are not covered, and the corresponding rotation-number comparison is not established. The $A_0\ge 0$00-pyramidal application is limited to $A_0\ge 0$01 by the same condition. For the hip-hop problem, although infinitely many relative periodic orbits exist on each compact energy surface, the winding number about the symmetry axis remains undetermined. Finally, the result concerns only compact regular energy surfaces in the window $A_0\ge 0$02; unbounded energy surfaces ($A_0\ge 0$03) fall outside the scope, though analogous infinite-orbit statements have been obtained for non-compact surfaces in related problems such as the spatial isosceles three-body problem.

Conclusion

The paper establishes that the Gutzwiller-type anisotropic Kepler problem admits infinitely many periodic orbits on every compact regular energy surface whenever the effective anisotropy parameter satisfies $A_0\ge 0$04, with equality in the governing volume–rotation-number inequality characterizing exactly the integrable spatial Kepler problem. The proof synthesizes Jensen-based contact volume comparison, Maslov-index computation of the Seifert rotation number through the Hill stability equation, and the CHHL two-or-infinity dichotomy. As applications, it yields infinitely many hip-hop-symmetric relative periodic orbits in the $A_0\ge 0$05-body problem for all $A_0\ge 0$06 and all central mass ratios, and infinitely many periodic orbits in the $A_0\ge 0$07-pyramidal problem for all mass ratios when $A_0\ge 0$08.