No infinite spin for planar total collision

Moeckel, Richard; Montgomery, Richard

doi:10.1090/jams/1044

No infinite spin for planar total collision
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by Richard Moeckel and Richard Montgomery;

J. Amer. Math. Soc.

DOI: https://doi.org/10.1090/jams/1044

Published electronically: January 18, 2024

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Abstract:

The infinite spin problem is an old problem concerning the rotational behavior of total collision orbits in the n-body problem. It has long been known that when a solution tends to total collision then its normalized configuration curve must converge to the set of normalized central configurations. In the planar n-body problem every normalized configuration determines a circle of rotationally equivalent normalized configurations and, in particular, there are circles of normalized central configurations. It’s conceivable that by means of an infinite spin, a total collision solution could converge to such a circle instead of to a particular point on it. Here we prove that this is not possible, at least if the limiting circle of central configurations is isolated from other circles of central configurations. (It is believed that all central configurations are isolated, but this is not known in general.) Our proof relies on combining the center manifold theorem with the Łojasiewicz gradient inequality.

References

Alain Albouy, Hildeberto E. Cabral, and Alan A. Santos, Some problems on the classical $n$-body problem, Celestial Mech. Dynam. Astronom. 113 (2012), no. 4, 369–375. MR 2970201, DOI 10.1007/s10569-012-9431-1
Alain Albouy and Vadim Kaloshin, Finiteness of central configurations of five bodies in the plane, Ann. of Math. (2) 176 (2012), no. 1, 535–588. MR 2925390, DOI 10.4007/annals.2012.176.1.10
A. Bressan, Tutorial on the center manifold theorem, Springer LNM 1911, Hyperbolic systems of balance laws (2003), pp. 327–344.
J. Chazy, Sur certaines trajectoires du problème des n corps, Bull. Astron. 35 (1918), 321–389.
A. Chenciner, Collisions totales, mouvements complètement paraboliques et réduction des homothéties dans le problème des $n$ corps, Regul. Chaotic Dyn. 3 (1998), no. 3, 93–106 (French, with French and Russian summaries). J. Moser at 70 (Russian). MR 1704972, DOI 10.1070/rd1998v003n03ABEH000083
Tobias Holck Colding and William P. Minicozzi II, Łojasiewicz inequalities and applications, Surveys in differential geometry 2014. Regularity and evolution of nonlinear equations, Surv. Differ. Geom., vol. 19, Int. Press, Somerville, MA, 2015, pp. 63–82. MR 3381496, DOI 10.4310/SDG.2014.v19.n1.a3
Mohamed Sami ElBialy, Collision singularities in celestial mechanics, SIAM J. Math. Anal. 21 (1990), no. 6, 1563–1593. MR 1075592, DOI 10.1137/0521086
N. Fenichel, Persistence and smoothness of invariant manifolds for flows, Indiana Univ. Math. J. 21 (1972), 193–226.
Marshall Hampton and Richard Moeckel, Finiteness of relative equilibria of the four-body problem, Invent. Math. 163 (2006), no. 2, 289–312. MR 2207019, DOI 10.1007/s00222-005-0461-0
S. Łojasiewicz, Sur les trajectoires du gradient d’une fonction analytique, Seminari di Geometria 1982-1983, Università di Bologna, (1984), 115–117.
Richard McGehee, Triple collision in the collinear three-body problem, Invent. Math. 27 (1974), 191–227. MR 359459, DOI 10.1007/BF01390175
R. McGehee, Singularities in classical celestial mechanics, Proc. of Int. Cong. Math., Helsinki (1978), 827–834.
Kenneth R. Meyer and Dieter S. Schmidt, Bifurcations of relative equilibria in the $4$- and $5$-body problem, Ergodic Theory Dynam. Systems 8$^*$ (1988), no. Charles Conley Memorial Issue, 215–225. MR 967639, DOI 10.1017/S0143385700009433
Richard Montgomery, A tour of subriemannian geometries, their geodesics and applications, Mathematical Surveys and Monographs, vol. 91, American Mathematical Society, Providence, RI, 2002. MR 1867362, DOI 10.1090/surv/091
R. Montgomery, Isoholonomic problems and some applications, Comm. Math. Phys. 128 (1990), no. 3, 565–592. MR 1045885, DOI 10.1007/BF02096874
Richard Montgomery, The three-body problem and the shape sphere, Amer. Math. Monthly 122 (2015), no. 4, 299–321. MR 3343067, DOI 10.4169/amer.math.monthly.122.04.299
Julian I. Palmore, Classifying relative equilibria. III, Lett. Math. Phys. 1 (1975/76), no. 1, 71–73. MR 413647, DOI 10.1007/BF00405589
Donald G. Saari and Neal D. Hulkower, On the manifolds of total collapse orbits and of completely parabolic orbits for the $n$-body problem, J. Differential Equations 41 (1981), no. 1, 27–43. MR 626619, DOI 10.1016/0022-0396(81)90051-6
Donald G. Saari, Symmetry in $n$-particle systems, Hamiltonian dynamical systems (Boulder, CO, 1987) Contemp. Math., vol. 81, Amer. Math. Soc., Providence, RI, 1988, pp. 23–42. MR 986255, DOI 10.1090/conm/081/986255
Steve Smale, Mathematical problems for the next century, Math. Intelligencer 20 (1998), no. 2, 7–15. MR 1631413, DOI 10.1007/BF03025291
Karl F. Sundman, Mémoire sur le problème des trois corps, Acta Math. 36 (1913), no. 1, 105–179 (French). MR 1555085, DOI 10.1007/BF02422379
Xiang Yu, On the problem of infinite spin in the total collision of the planar n-body problem, (2021), arXiv:1911.11006, DOI 10.48550/arXiv.1911.11006.
Stephen Wiggins, Normally hyperbolic invariant manifolds in dynamical systems, Applied Mathematical Sciences, vol. 105, Springer-Verlag, New York, 1994. With the assistance of György Haller and Igor Mezić. MR 1278264, DOI 10.1007/978-1-4612-4312-0
A. Wintner, The analytical foundations of celestial mechanics, Princeton University Press, Princeton, NJ, 1941.
Zhihong Xia, Central configurations with many small masses, J. Differential Equations 91 (1991), no. 1, 168–179. MR 1106123, DOI 10.1016/0022-0396(91)90137-X