Morse Theory for S-balanced Configurations in the Newtonian n-body Problem

Medientyp: E-Artikel
Titel: Morse Theory for S-balanced Configurations in the Newtonian n-body Problem
Beteiligte: Asselle, Luca; Portaluri, Alessandro
Erschienen: Springer Science and Business Media LLC, 2023
Erschienen in: Journal of Dynamics and Differential Equations
Sprache: Englisch
DOI: 10.1007/s10884-021-10036-y
ISSN: 1040-7294; 1572-9222
Schlagwörter: Analysis
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Beschreibung: <jats:title>Abstract</jats:title><jats:p>For the Newtonian (gravitational) <jats:italic>n</jats:italic>-body problem in the Euclidean <jats:italic>d</jats:italic>-dimensional space, the simplest possible solutions are provided by those rigid motions (homographic solutions) in which each body moves along a Keplerian orbit and the configuration of the <jats:italic>n</jats:italic>-body is a (constant up to rotations and scalings) <jats:italic>central configuration</jats:italic>. For <jats:inline-formula><jats:alternatives><jats:tex-math>$$d\le 3$$</jats:tex-math><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>d</mml:mi> <mml:mo>≤</mml:mo> <mml:mn>3</mml:mn> </mml:mrow> </mml:math></jats:alternatives></jats:inline-formula>, the only possible homographic motions are those given by central configurations. For <jats:inline-formula><jats:alternatives><jats:tex-math>$$d \ge 4$$</jats:tex-math><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>d</mml:mi> <mml:mo>≥</mml:mo> <mml:mn>4</mml:mn> </mml:mrow> </mml:math></jats:alternatives></jats:inline-formula> instead, new possibilities arise due to the higher complexity of the orthogonal group <jats:inline-formula><jats:alternatives><jats:tex-math>$$\mathrm {O}(d)$$</jats:tex-math><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>O</mml:mi> <mml:mo>(</mml:mo> <mml:mi>d</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> </mml:math></jats:alternatives></jats:inline-formula>, as observed by Albouy and Chenciner (Invent Math 131(1):151–184, 1998). For instance, in <jats:inline-formula><jats:alternatives><jats:tex-math>$$\mathbb {R}^4$$</jats:tex-math><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msup> <mml:mrow> <mml:mi>R</mml:mi> </mml:mrow> <mml:mn>4</mml:mn> </mml:msup> </mml:math></jats:alternatives></jats:inline-formula> it is possible to rotate in two mutually orthogonal planes with different angular velocities. This produces a new balance between gravitational forces and centrifugal forces providing new periodic and quasi-periodic motions. So, for <jats:inline-formula><jats:alternatives><jats:tex-math>$$d\ge 4$$</jats:tex-math><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>d</mml:mi> <mml:mo>≥</mml:mo> <mml:mn>4</mml:mn> </mml:mrow> </mml:math></jats:alternatives></jats:inline-formula> there is a wider class of <jats:italic>S</jats:italic><jats:italic>-balanced configurations</jats:italic> (containing the central ones) providing simple solutions of the <jats:italic>n</jats:italic>-body problem, which can be characterized as well through critical point theory. In this paper, we first provide a lower bound on the number of balanced (non-central) configurations in <jats:inline-formula><jats:alternatives><jats:tex-math>$$\mathbb {R}^d$$</jats:tex-math><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msup> <mml:mrow> <mml:mi>R</mml:mi> </mml:mrow> <mml:mi>d</mml:mi> </mml:msup> </mml:math></jats:alternatives></jats:inline-formula>, for arbitrary <jats:inline-formula><jats:alternatives><jats:tex-math>$$d\ge 4$$</jats:tex-math><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>d</mml:mi> <mml:mo>≥</mml:mo> <mml:mn>4</mml:mn> </mml:mrow> </mml:math></jats:alternatives></jats:inline-formula>, and establish a version of the <jats:inline-formula><jats:alternatives><jats:tex-math>$$45^\circ $$</jats:tex-math><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msup> <mml:mn>45</mml:mn> <mml:mo>∘</mml:mo> </mml:msup> </mml:math></jats:alternatives></jats:inline-formula>-theorem for balanced configurations, thus answering some of the questions raised in Moeckel (Central configurations, 2014). Also, a careful study of the asymptotics of the coefficients of the Poincaré polynomial of the collision free configuration sphere will enable us to derive some rather unexpected qualitative consequences on the count of <jats:italic>S</jats:italic>-balanced configurations. In the last part of the paper, we focus on the case <jats:inline-formula><jats:alternatives><jats:tex-math>$$d=4$$</jats:tex-math><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>d</mml:mi> <mml:mo>=</mml:mo> <mml:mn>4</mml:mn> </mml:mrow> </mml:math></jats:alternatives></jats:inline-formula> and provide a lower bound on the number of periodic and quasi-periodic motions of the gravitational <jats:italic>n</jats:italic>-body problem which improves a previous celebrated result of McCord (Ergodic Theory Dyn Syst 16:1059–1070, 1996).</jats:p>

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