Positive elliptic-elliptic rotopulsators on Clifford tori of nonconstant\n size project onto regular polygons

2020/03/24 by Pieter Tibboel, Tibboel, Pieter
Mathematics · Physics and Astronomy · #Algebraic and Geometric Analysis #Classical Analysis and ODEs (math.CA) #FOS: Mathematics #Holomorphic and Operator Theory #Quantum chaos and dynamical systems #Spectral Theory in Mathematical Physics

paper · pdf · doi:10.48550/arxiv.2003.11204

openalex publication_date 2020/03/24 · openalex created_date 2022/07/26 · openalex updated_date 2026/07/28

Abstract

Let q₁,...,q_n be the position vectors of the point masses of the\ncurved n-body problem. Consider any positive elliptic-elliptic rotopulsator\nsolution\nq_i^T=(r\cos(\θ+\α_i),r\sin(\θ+\α_i),\ρ\cos(\φ+\β_i),\ρ\sin(\φ+\β_i)),\ni\∈ 1,...,n , where \α₁,...,\α_n,\β₁,...,\β_n\∈\n[0,2\π) are constants, \φ, \θ, r and \ρ are\ntwice-differentiable, continuous, nonconstant functions, r²+\ρ²=1,\nr\≥ 0 and \ρ\≥ 0. We prove that the if the configuration of the point\nmasses is of nonconstant size, the configuration of the vectors\n (r\cos(\θ+\α_i),r\sin(\θ+\α_i))^T is a regular\npolygon, as is the configuration of the vectors\n(\ρ\cos(\φ+\β_i),\ρ\sin(\φ+\β_i))^T,\ni\∈ 1,...,n .\n

Citations

• On the non-existence of hyperbolic polygonal relative equilibria for the negative curved n--body problem with equal masses
• Choreography solutions of the n-body problem on S²

Related

• Classification of positive elliptic-elliptic rotopulsators on Clifford tori
• Polygonal negative hyperbolic rotopulsators of the curved n-body\n problem
• Circular non-collision orbits for a large class of n-body problems
• Existence of a class of rotopulsators
• Central configurations of the curved N-body problem