Superintegrability and choreographic obstructions in dihedral $n$-body Hamiltonian systems

TL;DR

This work analyzes planar $n$-body systems with dihedral symmetry $D_n$ and quadratic interactions, showing that while normal-mode frequencies can be commensurate (maximal superintegrability) periodicity occurs, choreography requires a sectorwise phase-matching condition across active Fourier sectors. The central result is a general phase-matching theorem: a $T$-periodic solution is $C_n$-equivariant if and only if every dynamically active Fourier sector satisfies $e^{i\Omega_\ell T/n}=e^{2\pi i\ell/n}$, which distinguishes choreography from mere periodicity. For $n\le 5$, phase-matching enforces a single geometric trace; for $n=6$ and higher, multiple active sectors generically yield multi-trace motions, with special degeneracies producing choreographic fragmentation or genuine multi-body choreographies in reduced subspaces. The paper provides explicit analyses for $n=4,5,6$, including the notable $1{:}2{:}3$ resonance at $n=6$ (typically non-choreographic) and the degenerate $1{:}2{:}2$ locking that yields a true six-body choreography; extension to arbitrary $n$ suggests fragmentation as the generic outcome. Overall, the study clarifies how symmetry, resonance, and representation theory jointly determine the existence or obstruction of planar choreographies in these symmetric $n$-body systems, offering a robust framework for identifying choreographies and their fragmentation across $n$.

Abstract

In this study $n$-body systems on the plane, invariant under the dihedral group $D_n$, with quadratic pairwise interactions are considered. In the center-of-mass frame the dynamics separates into Fourier normal modes. Although suitable couplings can make the system maximally superintegrable (so all bounded motions are periodic), resonance of the mode frequencies does not by itself ensure choreographic motion. We give an explicit criterion for choreographic symmetry of a $T$-periodic solution: invariance under the choreography generator (time shift $T/n$ combined with cyclic relabeling/rotation) holds if and only if every dynamically active Fourier sector satisfies a sectorwise phase condition. This yields a direct test for when a closed orbit can be realized as a single-trace $n$-body choreography. When several symmetry-distinct mode families are simultaneously excited, periodic motions are generically not single-trace; for generic superintegrable initial data the particles lie on distinct closed curves, producing multi-trace motion. In special cases this multi-trace structure organizes into sub-choreographies supported on distinct traces; we refer to this structured splitting as choreographic fragmentation. The cases $n=4,5,6$ are analyzed explicitly. In particular, for $n=6$ the maximally superintegrable resonance $1{:}2{:}3$ yields periodic but generically multi-trace dynamics, whereas a single-trace six-body choreography occurs only at the degenerate resonance $1{:}2{:}2$.

Figures (8)

• Figure 1: Distinguished relative coordinates and the action of $D_4$ for $n=4$. Here $\sigma_X$ and $\sigma_d$ denote reflections across axial and diagonal mirror axes of the square, respectively.
• Figure 2: Four-body $D_4$-invariant quadratic system in the superintegrable $(1\!:\!2)$ regime ($m=1$, $\omega=1$, $\kappa^{(4)}_1=1$, $\kappa^{(4)}_2=-\tfrac{1}{2}$) showing a $(2+2)$ choreographic fragmentation. Initial conditions at $t=0$: $\mathbf r_1=(1,1)$, $\mathbf r_2=(-1,\tfrac{1}{2})$, $\mathbf r_3=(0,0)$, $\mathbf r_4=(-\tfrac{1}{2},-\tfrac{1}{2})$; $\mathbf p_1=(0,\tfrac{3}{2})$, $\mathbf p_2=(-\tfrac{1}{2},-1)$, $\mathbf p_3=(0,\tfrac{1}{2})$, $\mathbf p_4=(\tfrac{1}{2},-1)$. Particles $(1,3)$ and $(2,4)$ form two synchronized dimers, each executing a two-body choreography with time shift $T/2$; the full motion is periodic but only $C_2$-equivariant (hence not a single four-body choreography).
• Figure 3: Four-body $(1\!:\!2)$ limaçon choreography in the superintegrable $D_4$-invariant quadratic model with $m=1$, $\omega=1$, $\kappa^{(4)}_1=1$, $\kappa^{(4)}_2=-\tfrac{1}{2}$ (hence $\Omega_{\mathrm F}=1$, $\Omega_{\mathrm N}=2$). Initial conditions at $t=0$: $\mathbf r_1=(1,0)$, $\mathbf r_2=(-\tfrac{1}{2},\tfrac{1}{2})$, $\mathbf r_3=(0,0)$, $\mathbf r_4=(-\tfrac{1}{2},-\tfrac{1}{2})$; $\mathbf p_1=(0,\tfrac{3}{2})$, $\mathbf p_2=(-\tfrac{1}{2},-1)$, $\mathbf p_3=(0,\tfrac{1}{2})$, $\mathbf p_4=(\tfrac{1}{2},-1)$. All four particles traverse the same closed curve with time shift $T/4$, realizing the primitive $1{:}2$ superintegrable four-body choreography.
• Figure 4: Second-neighbor relative vectors for $n=5$. Blue arrows represent the equivalent second-neighbor bonds. These variables diagonalize the quadratic $D_5$-invariant Hamiltonian.
• Figure 5: Five-body $D_5$–invariant quadratic system in the superintegrable $(1\!:\!2)$ regime ($m=1$, $\omega=1$, $\kappa^{(5)}_1=\tfrac{1}{2}(\tfrac{3}{\sqrt5}+1)$, $\kappa^{(5)}_2=-\tfrac{1}{2}(\tfrac{3}{\sqrt5}-1)$) showing a $(2+2+1)$ choreographic fragmentation. Initial conditions at $t=0$: $\mathbf r_1=(-0.144427191,\,0.079179607)$, $\mathbf r_2=(0.659016994,\,0.140983006)$, $\mathbf r_3=(1.0,\,0.5)$, $\mathbf r_4=(0.2,\,-0.1)$, $\mathbf r_5=(-0.3,\,0.4)$; $\mathbf p_1=(0.1,\,-0.090450850)$, $\mathbf p_2=(0.023606798,\,-0.002254249)$, $\mathbf p_3=(-0.223606798,\,0.192705098)$, $\mathbf p_4=(0.2,\,-0.15)$, $\mathbf p_5=(-0.1,\,0.05)$. Two synchronized dimers and one isolated particle evolve on distinct closed curves; the motion is periodic but only equivariant under a proper cyclic subgroup of $C_5$, and therefore does not form a single five-body choreography (choreographic fragmentation).

Theorems & Definitions (10)

• Remark 1: Normalization for even $n$
• Proposition 1: Normal--mode decomposition
• proof : Proof
• Theorem 2: $C_n$-equivariance (phase-matching) criterion
• proof
• Remark 2: Equivariance vs. single-trace choreography
• Proposition 3: Generic obstruction to single--trace under multi-sector excitation
• proof
• Theorem 4: Obstruction to $C_n$-equivariance (and hence choreography)
• proof