Viscosity solutions and hyperbolic motions: a new PDE method for the $N$-body problem

Abstract

We prove for the $N$-body problem the existence of hyperbolic motions for any prescribed limit shape and any given initial configuration of the bodies. The energy level $h>0$ of the motion can also be chosen arbitrarily. Our approach is based on the construction of global viscosity solutions for the Hamilton-Jacobi equation $H(x,d_xu)=h$. We prove that these solutions are fixed points of the associated Lax-Oleinik semigroup. The presented results can also be viewed as a new application of Marchal’s Theorem, whose main use in recent literature has been to prove the existence of periodic orbits.

  • [Alb] A. Albouy, "Lectures on the two-body problem," in Classical and Celestial Mechanics, Princeton Univ. Press, Princeton, NJ, 2002, pp. 63-116.
    @INCOLLECTION{Alb,
      author = {Albouy, Alain},
      title = {Lectures on the two-body problem},
      booktitle = {Classical and Celestial Mechanics},
      note = {H. Cabral, F. Diacu, editors},
      venue = {{R}ecife, 1993/1999},
      pages = {63--116},
      publisher = {Princeton Univ. Press, Princeton, NJ},
      year = {2002},
      mrclass = {70F05 (70F15)},
      mrnumber = {1974780},
      mrreviewer = {Jes\'{u}s F. Palaci\'{a}n},
      zblnumber = {1181.70014},
      }
  • [AlbKal] Go to document A. Albouy and V. Kaloshin, "Finiteness of central configurations of five bodies in the plane," Ann. of Math. (2), vol. 176, iss. 1, pp. 535-588, 2012.
    @ARTICLE{AlbKal,
      author = {Albouy, Alain and Kaloshin, Vadim},
      title = {Finiteness of central configurations of five bodies in the plane},
      journal = {Ann. of Math. (2)},
      fjournal = {Annals of Mathematics. Second Series},
      volume = {176},
      year = {2012},
      number = {1},
      pages = {535--588},
      issn = {0003-486X},
      mrclass = {70F10 (05C62 32C25 70G10)},
      mrnumber = {2925390},
      mrreviewer = {Josep M. Cors},
      doi = {10.4007/annals.2012.176.1.10},
      url = {https://doi.org/10.4007/annals.2012.176.1.10},
      zblnumber = {1362.70014},
      }
  • [ArPeChSt] Go to document J. A. Arredondo, E. Pérez-Chavela, and C. Stoica, "Dynamics in the Schwarzschild isosceles three body problem," J. Nonlinear Sci., vol. 24, iss. 6, pp. 997-1032, 2014.
    @ARTICLE{ArPeChSt,
      author = {Arredondo, John A. and Pérez-Chavela, Ernesto and Stoica, Cristina},
      title = {Dynamics in the {S}chwarzschild isosceles three body problem},
      journal = {J. Nonlinear Sci.},
      fjournal = {Journal of Nonlinear Science},
      volume = {24},
      year = {2014},
      number = {6},
      pages = {997--1032},
      issn = {0938-8974},
      mrclass = {70F07 (70F15 83C10)},
      mrnumber = {3275217},
      mrreviewer = {Andrzej M. Frydryszak},
      doi = {10.1007/s00332-014-9210-0},
      url = {https://doi.org/10.1007/s00332-014-9210-0},
      zblnumber = {1305.70019},
      }
  • [BaGrSch] Go to document W. Ballmann, M. Gromov, and V. Schroeder, Manifolds of Nonpositive Curvature, Birkhäuser Boston, Inc., Boston, MA, 1985, vol. 61.
    @BOOK{BaGrSch,
      author = {Ballmann, Werner and Gromov, Mikhael and Schroeder, Viktor},
      title = {Manifolds of Nonpositive Curvature},
      series = {Progr. Math.},
      volume = {61},
      publisher = {Birkhäuser Boston, Inc., Boston, MA},
      year = {1985},
      pages = {vi+263},
      isbn = {0-8176-3181-X},
      mrclass = {53C20},
      mrnumber = {0823981},
      mrreviewer = {Gudlaugur Thorbergsson},
      doi = {10.1007/978-1-4684-9159-3},
      url = {https://doi.org/10.1007/978-1-4684-9159-3},
      zblnumber = {0591.53001},
      }
  • [BarCap] Go to document M. Bardi and I. Capuzzo-Dolcetta, Optimal Control and Viscosity Solutions of Hamilton-Jacobi-Bellman Equations, Birkhäuser Boston, Inc., Boston, MA, 1997.
    @BOOK{BarCap,
      author = {Bardi, Martino and Capuzzo-Dolcetta, Italo},
      title = {Optimal Control and Viscosity Solutions of {H}amilton-{J}acobi-{B}ellman Equations},
      series = {Systems Control Found. Appl.},
      note = {with appendices by Maurizio Falcone and Pierpaolo Soravia},
      publisher = {Birkhäuser Boston, Inc., Boston, MA},
      year = {1997},
      pages = {xviii+570},
      isbn = {0-8176-3640-4},
      mrclass = {49-02 (49J15 49K15 49L25)},
      mrnumber = {1484411},
      mrreviewer = {Vladimir Veliov},
      doi = {10.1007/978-0-8176-4755-1},
      url = {https://doi.org/10.1007/978-0-8176-4755-1},
      zblnumber = {0890.49011},
      }
  • [Bar] G. Barles, Solutions de Viscosité des Équations de Hamilton-Jacobi, Springer-Verlag, Paris, 1994, vol. 17.
    @BOOK{Bar,
      author = {Barles, Guy},
      title = {Solutions de Viscosité des \'{E}quations de {H}amilton-{J}acobi},
      series = {Math. Appl. (Berlin)},
      volume = {17},
      publisher = {Springer-Verlag, Paris},
      year = {1994},
      pages = {x+194},
      isbn = {3-540-58422-6},
      mrclass = {49L25 (35D05 35F20 35J60)},
      mrnumber = {1613876},
      mrreviewer = {Martino Bardi},
      zblnumber = {0819.35002},
      }
  • [Ber] Go to document P. Bernard, "The Lax-Oleinik semi-group: a Hamiltonian point of view," Proc. Roy. Soc. Edinburgh Sect. A, vol. 142, iss. 6, pp. 1131-1177, 2012.
    @ARTICLE{Ber,
      author = {Bernard, Patrick},
      title = {The {L}ax-{O}leinik semi-group: a {H}amiltonian point of view},
      journal = {Proc. Roy. Soc. Edinburgh Sect. A},
      fjournal = {Proceedings of the Royal Society of Edinburgh. Section A. Mathematics},
      volume = {142},
      year = {2012},
      number = {6},
      pages = {1131--1177},
      issn = {0308-2105},
      mrclass = {70H08 (37J40 70H20)},
      mrnumber = {3002592},
      mrreviewer = {José Claudio Vidal Diaz},
      doi = {10.1017/S0308210511000059},
      url = {https://doi.org/10.1017/S0308210511000059},
      zblnumber = {1400.70027},
      }
  • [Cha1] J. Chazy, "Sur certaines trajectoires du probléme des $n$ corps," Bull. Astronom., vol. 35, pp. 321-389, 1918.
    @ARTICLE{Cha1,
      author = {Chazy, J.},
      title = {Sur certaines trajectoires du probléme des $n$ corps},
      journal = {Bull. Astronom.},
      volume = {35},
      year = {1918},
      pages = {321--389},
      zblnumber = {},
      }
  • [Cha2] Go to document J. Chazy, "Sur l’allure du mouvement dans le problème des trois corps quand le temps cro\^ıt indéfiniment," Ann. Sci. École Norm. Sup. (3), vol. 39, pp. 29-130, 1922.
    @ARTICLE{Cha2,
      author = {Chazy, Jean},
      title = {Sur l'allure du mouvement dans le problème des trois corps quand le temps cro\^ıt indéfiniment},
      journal = {Ann. Sci. \'{E}cole Norm. Sup. (3)},
      fjournal = {Annales Scientifiques de l'\'{E}cole Normale Supérieure. Troisième Série},
      volume = {39},
      year = {1922},
      pages = {29--130},
      issn = {0012-9593},
      mrclass = {DML},
      mrnumber = {1509241},
      doi = {10.24033/asens.739},
      url = {https://doi.org/10.24033/asens.739},
      jfmnumber = {48.1074.04},
      }
  • [Che2] Go to document A. Chenciner, "À l’infini en temps fini," in Séminaire Bourbaki, Vol. 1996/97, Soc. Math. France, Paris, 1997, p. exp. no. 832, 5, 323-353.
    @INCOLLECTION{Che2,
      author = {Chenciner, Alain},
      title = {À l'infini en temps fini},
      booktitle = {Séminaire Bourbaki, Vol. 1996/97},
      series = {Astérisque},
      fjournal = {Astérisque},
      publisher = {Soc. Math. France, Paris},
      number = {245},
      year = {1997},
      pages = {Exp. No. 832, 5, 323--353},
      issn = {0303-1179},
      mrclass = {70F10},
      mrnumber = {1627117},
      mrreviewer = {Joseph L. Gerver},
      zblnumber = {0930.70011},
      url = {http://www.numdam.org/item/SB_1996-1997__39__323_0/},
      }
  • [Che3] Go to document A. Chenciner, "Collisions totales, mouvements complètement paraboliques et réduction des homothéties dans le problème des $n$ corps," Regul. Chaotic Dyn., vol. 3, iss. 3, pp. 93-106, 1998.
    @ARTICLE{Che3,
      author = {Chenciner, Alain},
      title = {Collisions totales, mouvements complètement paraboliques et réduction des homothéties dans le problème des {$n$} corps},
      note = {J. Moser at 70 (Russian)},
      journal = {Regul. Chaotic Dyn.},
      fjournal = {Regular \& Chaotic Dynamics. Regulyarnaya \& Khaoticheskaya Dinamika},
      volume = {3},
      year = {1998},
      number = {3},
      pages = {93--106},
      issn = {1560-3547},
      mrclass = {70F10 (70F15 70F16)},
      mrnumber = {1704972},
      mrreviewer = {Florin N. Diacu},
      doi = {10.1070/rd1998v003n03ABEH000083},
      url = {https://doi.org/10.1070/rd1998v003n03ABEH000083},
      zblnumber = {0973.70011},
      }
  • [Che1] Go to document A. Chenciner, "Action minimizing solutions of the Newtonian $n$-body problem: from homology to symmetry," in Proceedings of the International Congress of Mathematicians, Vol. III, 2002, pp. 279-294.
    @INPROCEEDINGS{Che1,
      author = {Chenciner, Alain},
      title = {Action minimizing solutions of the {N}ewtonian {$n$}-body problem: from homology to symmetry},
      booktitle = {Proceedings of the {I}nternational {C}ongress of {M}athematicians, {V}ol. {III}},
      venue = {{B}eijing, 2002},
      pages = {279--294},
      publisher = {Higher Ed. Press, Beijing},
      year = {2002},
      mrclass = {70F07 (37J45 70F10 70F16)},
      mrnumber = {1957539},
      mrreviewer = {Joseph L. Gerver},
      zblnumber = {1136.70310},
      url = {https://www.mathunion.org/fileadmin/ICM/Proceedings/ICM2002.3/ICM2002.3.ocr.pdf},
      }
  • [CheMon] Go to document A. Chenciner and R. Montgomery, "A remarkable periodic solution of the three-body problem in the case of equal masses," Ann. of Math. (2), vol. 152, iss. 3, pp. 881-901, 2000.
    @ARTICLE{CheMon,
      author = {Chenciner, Alain and Montgomery, Richard},
      title = {A remarkable periodic solution of the three-body problem in the case of equal masses},
      journal = {Ann. of Math. (2)},
      fjournal = {Annals of Mathematics. Second Series},
      volume = {152},
      year = {2000},
      number = {3},
      pages = {881--901},
      issn = {0003-486X},
      mrclass = {70F07 (37J50)},
      mrnumber = {1815704},
      mrreviewer = {Florin N. Diacu},
      doi = {10.2307/2661357},
      url = {https://doi.org/10.2307/2661357},
      zblnumber = {0987.70009},
      }
  • [CraLio] Go to document M. G. Crandall and P. Lions, "Viscosity solutions of Hamilton-Jacobi equations," Trans. Amer. Math. Soc., vol. 277, iss. 1, pp. 1-42, 1983.
    @ARTICLE{CraLio,
      author = {Crandall, Michael G. and Lions, Pierre-Louis},
      title = {Viscosity solutions of {H}amilton-{J}acobi equations},
      journal = {Trans. Amer. Math. Soc.},
      fjournal = {Transactions of the Amer. Math. Soc.},
      volume = {277},
      year = {1983},
      number = {1},
      pages = {1--42},
      issn = {0002-9947},
      mrclass = {35F20},
      mrnumber = {0690039},
      mrreviewer = {Moshe Marcus},
      doi = {10.2307/1999343},
      url = {https://doi.org/10.2307/1999343},
      zblnumber = {0599.35024},
      }
  • [CraEvaLio] Go to document M. G. Crandall, L. C. Evans, and P. -L. Lions, "Some properties of viscosity solutions of Hamilton-Jacobi equations," Trans. Amer. Math. Soc., vol. 282, iss. 2, pp. 487-502, 1984.
    @ARTICLE{CraEvaLio,
      author = {Crandall, M. G. and Evans, L. C. and Lions, P.-L.},
      title = {Some properties of viscosity solutions of {H}amilton-{J}acobi equations},
      journal = {Trans. Amer. Math. Soc.},
      fjournal = {Transactions of the Amer. Math. Soc.},
      volume = {282},
      year = {1984},
      number = {2},
      pages = {487--502},
      issn = {0002-9947},
      mrclass = {35F20 (35L60)},
      mrnumber = {0732102},
      doi = {10.2307/1999247},
      url = {https://doi.org/10.2307/1999247},
      zblnumber = {0543.35011},
      }
  • [daLMad] Go to document A. da Luz and E. Maderna, "On the free time minimizers of the Newtonian $N$-body problem," Math. Proc. Cambridge Philos. Soc., vol. 156, iss. 2, pp. 209-227, 2014.
    @ARTICLE{daLMad,
      author = {da Luz, Adriana and Maderna, Ezequiel},
      title = {On the free time minimizers of the {N}ewtonian {$N$}-body problem},
      journal = {Math. Proc. Cambridge Philos. Soc.},
      fjournal = {Mathematical Proceedings of the Cambridge Philosophical Society},
      volume = {156},
      year = {2014},
      number = {2},
      pages = {209--227},
      issn = {0305-0041},
      mrclass = {70F10 (49K05 58E30 70H08)},
      mrnumber = {3177865},
      mrreviewer = {Diogo Pinheiro},
      doi = {10.1017/S0305004113000650},
      url = {https://doi.org/10.1017/S0305004113000650},
      zblnumber = {1331.70035},
      }
  • [Dia] F. Diacu, "Singularities of the $N$-body problem," in Classical and Celestial Mechanics, Princeton Univ. Press, Princeton, NJ, 2002, pp. 35-62.
    @INCOLLECTION{Dia,
      author = {Diacu, Florin},
      title = {Singularities of the {$N$}-body problem},
      booktitle = {Classical and Celestial Mechanics},
      venue = {{R}ecife, 1993/1999},
      pages = {35--62},
      publisher = {Princeton Univ. Press, Princeton, NJ},
      year = {2002},
      mrclass = {70F10 (70F15 70F16)},
      mrnumber = {1974779},
      mrreviewer = {Gareth E. Roberts},
      zblnumber = {1190.70007},
      }
  • [DuMoMoYu] Go to document N. Duignan, R. Moeckel, R. Montgomery, and G. Yu, "Chazy-type asymptotics and hyperbolic scattering for the $n$-body problem," Arch. Ration. Mech. Anal., vol. 238, iss. 1, pp. 255-297, 2020.
    @article{DuMoMoYu,
      author = {Duignan, Nathan and Moeckel, Richard and Montgomery, Richard and Yu, Guowei},
      TITLE = {Chazy-type asymptotics and hyperbolic scattering for the {$n$}-body problem},
      JOURNAL = {Arch. Ration. Mech. Anal.},
      FJOURNAL = {Archive for Rational Mechanics and Analysis},
      VOLUME = {238},
      YEAR = {2020},
      NUMBER = {1},
      PAGES = {255--297},
      ISSN = {0003-9527},
      MRCLASS = {70F10 (37N05)},
      MRNUMBER = {4121133},
      DOI = {10.1007/s00205-020-01542-2},
      URL = {https://doi.org/10.1007/s00205-020-01542-2},
      ZBLNUMBER = {07219763},
      }
  • [Eva1] Go to document L. C. Evans, Partial Differential Equations, Second ed., Amer. Math. Soc., Providence, RI, 2010, vol. 19.
    @BOOK{Eva1,
      author = {Evans, Lawrence C.},
      title = {Partial {D}ifferential {E}quations},
      series = {Grad. Stud. Math.},
      volume = {19},
      edition = {Second},
      publisher = {Amer. Math. Soc., Providence, RI},
      year = {2010},
      pages = {xxii+749},
      isbn = {978-0-8218-4974-3},
      mrclass = {35-01},
      mrnumber = {2597943},
      mrreviewer = {Diego M. Maldonado},
      doi = {10.1090/gsm/019},
      url = {https://doi.org/10.1090/gsm/019},
      zblnumber = {1194.35001},
      }
  • [Eva2] Go to document B. Dacorogna, "Calculus of variations, implicit partial differential equations and microstructure," GAMM-Mitt., vol. 29, iss. 2, pp. 150-171, 2006.
    @ARTICLE{Eva2,
      author = {Dacorogna, Bernard},
      title = {Calculus of variations, implicit partial differential equations and microstructure},
      journal = {GAMM-Mitt.},
      fjournal = {GAMM-Mitteilungen},
      volume = {29},
      year = {2006},
      number = {2},
      pages = {150--171},
      issn = {0936-7195},
      mrclass = {49J45 (35F25 35J20 74B20 74N15 74P05)},
      mrnumber = {2268764},
      doi = {10.1002/gamm.201490028},
      url = {https://doi.org/10.1002/gamm.201490028},
      zblnumber = {1157.49003},
      }
  • [Fat] Go to document A. Fathi, "Weak KAM theory: the connection between Aubry-Mather theory and viscosity solutions of the Hamilton-Jacobi equation," in Proceedings of the International Congress of Mathematicians—Seoul 2014. Vol. III, 2014, pp. 597-621.
    @INPROCEEDINGS{Fat,
      author = {Fathi, Albert},
      title = {Weak {KAM} theory: the connection between {A}ubry-{M}ather theory and viscosity solutions of the {H}amilton-{J}acobi equation},
      booktitle = {Proceedings of the {I}nternational {C}ongress of {M}athematicians---{S}eoul 2014. {V}ol. {III}},
      pages = {597--621},
      publisher = {Kyung Moon Sa, Seoul},
      year = {2014},
      mrclass = {37J50 (35D40 35F21 70H20)},
      mrnumber = {3729043},
      zblnumber = {1373.37151},
      url = {http://www.icm2014.org/download/Proceedings_Volume_III.pdf},
      }
  • [FerTer] Go to document D. L. Ferrario and S. Terracini, "On the existence of collisionless equivariant minimizers for the classical $n$-body problem," Invent. Math., vol. 155, iss. 2, pp. 305-362, 2004.
    @ARTICLE{FerTer,
      author = {Ferrario, Davide L. and Terracini, Susanna},
      title = {On the existence of collisionless equivariant minimizers for the classical {$n$}-body problem},
      journal = {Invent. Math.},
      fjournal = {Inventiones Mathematicae},
      volume = {155},
      year = {2004},
      number = {2},
      pages = {305--362},
      issn = {0020-9910},
      mrclass = {70F10 (37J45 49J40 49S05 70F07 70F16 70H30)},
      mrnumber = {2031430},
      mrreviewer = {Kuo-Chang Chen},
      doi = {10.1007/s00222-003-0322-7},
      url = {https://doi.org/10.1007/s00222-003-0322-7},
      zblnumber = {1068.70013},
      }
  • [Gro] Go to document M. Gromov, "Hyperbolic manifolds, groups and actions," in Riemann Surfaces and Related Topics: Proceedings of the 1978 Stony Brook Conference, 1981, pp. 183-213.
    @INPROCEEDINGS{Gro,
      author = {Gromov, M.},
      title = {Hyperbolic manifolds, groups and actions},
      booktitle = {Riemann Surfaces and Related Topics: {P}roceedings of the 1978 {S}tony {B}rook {C}onference},
      venue = {{S}tate {U}niv. {N}ew {Y}ork, {S}tony {B}rook, {N}.{Y}., 1978},
      series = {Ann. of Math. Stud.},
      volume = {97},
      pages = {183--213},
      publisher = {Princeton Univ. Press, Princeton, N.J.},
      year = {1981},
      mrclass = {53C15 (53C45 58F17)},
      mrnumber = {0624814},
      mrreviewer = {M. Rees},
      doi = {10.1515/9781400881550},
      url = {https://doi.org/10.1515/9781400881550},
      zblnumber = {0467.53035},
      }
  • [HamMoe] Go to document M. Hampton and R. Moeckel, "Finiteness of relative equilibria of the four-body problem," Invent. Math., vol. 163, iss. 2, pp. 289-312, 2006.
    @ARTICLE{HamMoe,
      author = {Hampton, Marshall and Moeckel, Richard},
      title = {Finiteness of relative equilibria of the four-body problem},
      journal = {Invent. Math.},
      fjournal = {Inventiones Mathematicae},
      volume = {163},
      year = {2006},
      number = {2},
      pages = {289--312},
      issn = {0020-9910},
      mrclass = {70F15 (37N05 70F10)},
      mrnumber = {2207019},
      mrreviewer = {Manuele Santoprete},
      doi = {10.1007/s00222-005-0461-0},
      url = {https://doi.org/10.1007/s00222-005-0461-0},
      zblnumber = {1083.70012},
      }
  • [Mad1] Go to document E. Maderna, "On weak KAM theory for $N$-body problems," Ergodic Theory Dynam. Systems, vol. 32, iss. 3, pp. 1019-1041, 2012.
    @ARTICLE{Mad1,
      author = {Maderna, Ezequiel},
      title = {On weak {KAM} theory for {$N$}-body problems},
      journal = {Ergodic Theory Dynam. Systems},
      fjournal = {Ergodic Theory and Dynamical Systems},
      volume = {32},
      year = {2012},
      number = {3},
      pages = {1019--1041},
      issn = {0143-3857},
      mrclass = {37J50 (31B15 70F10 70H20)},
      mrnumber = {2995654},
      mrreviewer = {Mikhail B. Sevryuk},
      doi = {10.1017/S0143385711000046},
      url = {https://doi.org/10.1017/S0143385711000046},
      zblnumber = {1257.37039},
      }
  • [Mad2] Go to document E. Maderna, "Minimizing configurations and Hamilton-Jacobi equations of homogeneous $N$-body problems," Regul. Chaotic Dyn., vol. 18, iss. 6, pp. 656-673, 2013.
    @ARTICLE{Mad2,
      author = {Maderna, Ezequiel},
      title = {Minimizing configurations and {H}amilton-{J}acobi equations of homogeneous {$N$}-body problems},
      journal = {Regul. Chaotic Dyn.},
      fjournal = {Regular and Chaotic Dynamics. International Scientific Journal},
      volume = {18},
      year = {2013},
      number = {6},
      pages = {656--673},
      issn = {1560-3547},
      mrclass = {70F10 (34C40 35D30 35D40 70H08 70H20)},
      mrnumber = {3146584},
      mrreviewer = {Hasna Riahi},
      doi = {10.1134/S1560354713060063},
      url = {https://doi.org/10.1134/S1560354713060063},
      zblnumber = {1286.70017},
      }
  • [MadVen] Go to document E. Maderna and A. Venturelli, "Globally minimizing parabolic motions in the Newtonian $N$-body problem," Arch. Ration. Mech. Anal., vol. 194, iss. 1, pp. 283-313, 2009.
    @ARTICLE{MadVen,
      author = {Maderna, Ezequiel and Venturelli, A.},
      title = {Globally minimizing parabolic motions in the {N}ewtonian {$N$}-body problem},
      journal = {Arch. Ration. Mech. Anal.},
      fjournal = {Archive for Rational Mechanics and Analysis},
      volume = {194},
      year = {2009},
      number = {1},
      pages = {283--313},
      issn = {0003-9527},
      mrclass = {74F10 (37J45 47J30)},
      mrnumber = {2533929},
      mrreviewer = {Zhifu Xie},
      doi = {10.1007/s00205-008-0175-8},
      url = {https://doi.org/10.1007/s00205-008-0175-8},
      zblnumber = {1253.70015},
      }
  • [Mar] Go to document C. Marchal, "How the method of minimization of action avoids singularities," Celestial Mech. Dynam. Astronom., vol. 83, iss. 1-4, pp. 325-353, 2002.
    @ARTICLE{Mar,
      author = {Marchal, C.},
      title = {How the method of minimization of action avoids singularities},
      note = {Modern celestial mechanics: from theory to applications (Rome, 2001)},
      journal = {Celestial Mech. Dynam. Astronom.},
      fjournal = {Celestial Mechanics \& Dynamical Astronomy. An International Journal of Space Dynamics},
      volume = {83},
      year = {2002},
      number = {1-4},
      pages = {325--353},
      issn = {0923-2958},
      mrclass = {70F10 (34H05 49K15 70H30)},
      mrnumber = {1956531},
      mrreviewer = {César Castilho},
      doi = {10.1023/A:1020128408706},
      url = {https://doi.org/10.1023/A:1020128408706},
      zblnumber = {1073.70011},
      }
  • [MarSaa] Go to document C. Marchal and D. G. Saari, "On the final evolution of the $n$-body problem," J. Differential Equations, vol. 20, iss. 1, pp. 150-186, 1976.
    @ARTICLE{MarSaa,
      author = {Marchal, Christian and Saari, Donald G.},
      title = {On the final evolution of the {$n$}-body problem},
      journal = {J. Differential Equations},
      fjournal = {Journal of Differential Equations},
      volume = {20},
      year = {1976},
      number = {1},
      pages = {150--186},
      issn = {0022-0396},
      mrclass = {70.34 (85.34)},
      mrnumber = {0416150},
      mrreviewer = {K. Forster},
      doi = {10.1016/0022-0396(76)90101-7},
      url = {https://doi.org/10.1016/0022-0396(76)90101-7},
      zblnumber = {0336.70010},
      }
  • [McG] R. McGehee, "Von Zeipel’s theorem on singularities in celestial mechanics," Exposition. Math., vol. 4, iss. 4, pp. 335-345, 1986.
    @ARTICLE{McG,
      author = {McGehee, Richard},
      title = {von {Z}eipel's theorem on singularities in celestial mechanics},
      journal = {Exposition. Math.},
      fjournal = {Expositiones Mathematicae. International Journal for Pure and Applied Mathematics},
      volume = {4},
      year = {1986},
      number = {4},
      pages = {335--345},
      issn = {0723-0869},
      mrclass = {70F10 (01A60 58F05 58F40)},
      mrnumber = {0867962},
      mrreviewer = {V. Szebehely},
      zblnumber = {0622.70005},
      }
  • [MoMoSa] Go to document R. Moeckel, R. Montgomery, and H. Sánchez Morgado, "Free time minimizers for the three-body problem," Celestial Mech. Dynam. Astronom., vol. 130, iss. 3, p. 28, 2018.
    @ARTICLE{MoMoSa,
      author = {Moeckel, Richard and Montgomery, Richard and S\'{a}nchez Morgado, Héctor},
      title = {Free time minimizers for the three-body problem},
      journal = {Celestial Mech. Dynam. Astronom.},
      fjournal = {Celestial Mechanics \& Dynamical Astronomy. An International Journal of Space Dynamics},
      volume = {130},
      year = {2018},
      number = {3},
      pages = {Paper No. 28, 28},
      issn = {0923-2958},
      mrclass = {70F10 (37N05 70F07 70F15 70G40 70G60)},
      mrnumber = {3779036},
      mrreviewer = {Nicola Soave},
      doi = {10.1007/s10569-018-9823-y},
      url = {https://doi.org/10.1007/s10569-018-9823-y},
      zblnumber = {1390.70022},
      }
  • [Mos] Go to document J. Moser, Stable and Random Motions in Dynamical Systems, Princeton Univ. Press, Princeton, N. J.; Univ. of Tokyo Press, Tokyo, 1973, vol. 77.
    @BOOK{Mos,
      author = {Moser, Jürgen},
      title = {Stable and Random Motions in Dynamical Systems},
      titlenote = {Hermann Weyl Lectures, the Institute for Advanced Study, Princeton, N. J},
      series = {Ann. of Math. Stud.},
      volume = {77},
      publisher = {Princeton Univ. Press, Princeton, N. J.; Univ. of Tokyo Press, Tokyo},
      year = {1973},
      pages = {viii+198},
      mrclass = {58FXX (34C35 70.58)},
      mrnumber = {0442980},
      mrreviewer = {Clark Robinson},
      zblnumber = {0271.70009},
      doi = {10.1515/9781400882694},
      url = {https://doi.org/10.1515/9781400882694},
      }
  • [Pal] Go to document J. I. Palmore, "Measure of degenerate relative equilibria. I," Ann. of Math. (2), vol. 104, iss. 3, pp. 421-429, 1976.
    @ARTICLE{Pal,
      author = {Palmore, Julian I.},
      title = {Measure of degenerate relative equilibria. {I}},
      journal = {Ann. of Math. (2)},
      fjournal = {Annals of Mathematics. Second Series},
      volume = {104},
      year = {1976},
      number = {3},
      pages = {421--429},
      issn = {0003-486X},
      mrclass = {58F05 (58E05 70.58)},
      mrnumber = {0420713},
      mrreviewer = {Donald G. Saari},
      doi = {10.2307/1970964},
      url = {https://doi.org/10.2307/1970964},
      zblnumber = {0321.58014},
      }
  • [Per] Go to document B. A. Percino-Figueroa, "Null angular momentum and weak KAM solutions of the Newtonian $N$-body problem," SIGMA Symmetry Integrability Geom. Methods Appl., vol. 13, p. 068, 2017.
    @ARTICLE{Per,
      author = {Percino-Figueroa, Boris A.},
      title = {Null angular momentum and weak {KAM} solutions of the {N}ewtonian {$N$}-body problem},
      journal = {SIGMA Symmetry Integrability Geom. Methods Appl.},
      fjournal = {SIGMA. Symmetry, Integrability and Geometry. Methods and Applications},
      volume = {13},
      year = {2017},
      pages = {Paper No. 068, 8},
      mrclass = {37J15 (37J50 70F10 70H20)},
      mrnumber = {3689149},
      mrreviewer = {A. S. Sumbatov},
      doi = {10.3842/SIGMA.2017.068},
      url = {https://doi.org/10.3842/SIGMA.2017.068},
      zblnumber = {1383.37050},
      }
  • [PerSan] Go to document B. Percino and H. Sánchez-Morgado, "Busemann functions for the $N$-body problem," Arch. Ration. Mech. Anal., vol. 213, iss. 3, pp. 981-991, 2014.
    @ARTICLE{PerSan,
      author = {Percino, Boris and S\'{a}nchez-Morgado, Héctor},
      title = {Busemann functions for the {$N$}-body problem},
      journal = {Arch. Ration. Mech. Anal.},
      fjournal = {Archive for Rational Mechanics and Analysis},
      volume = {213},
      year = {2014},
      number = {3},
      pages = {981--991},
      issn = {0003-9527},
      mrclass = {70F10 (49L25 70H08)},
      mrnumber = {3218835},
      mrreviewer = {Ezequiel Maderna},
      doi = {10.1007/s00205-014-0748-7},
      url = {https://doi.org/10.1007/s00205-014-0748-7},
      zblnumber = {1342.70031},
      }
  • [Poi] H. Poincaré, "Sur les solutions périodiques et le principe de moindre action," C. R. Acad. Sci., Paris, Sér. I, Math, vol. 123, pp. 915-918, 1896.
    @ARTICLE{Poi,
      author = {Poincaré, H.},
      title = {Sur les solutions périodiques et le principe de moindre action},
      journal = {C. R. Acad. Sci., Paris, Sér. I, Math},
      volume = {123},
      year = {1896},
      pages = {915--918},
      zblnumber = {27.0608.02},
      }
  • [Pol] Go to document H. Pollard, "The behavior of gravitational systems," J. Math. Mech., vol. 17, pp. 601-611, 1967/1968.
    @ARTICLE{Pol,
      author = {Pollard, Harry},
      title = {The behavior of gravitational systems},
      journal = {J. Math. Mech.},
      volume = {17},
      year = {1967/1968},
      pages = {601--611},
      mrclass = {70.34},
      mrnumber = {0261826},
      mrreviewer = {F. Nahon},
      doi = {10.1512/iumj.1968.17.17036},
      url = {https://doi.org/10.1512/iumj.1968.17.17036},
      zblnumber = {0159.26102},
      }
  • [SaaXia] Go to document D. G. Saari and Z. Xia, "The existence of oscillatory and superhyperbolic motion in Newtonian systems," J. Differential Equations, vol. 82, iss. 2, pp. 342-355, 1989.
    @ARTICLE{SaaXia,
      author = {Saari, Donald G. and Xia, Zhihong},
      title = {The existence of oscillatory and superhyperbolic motion in {N}ewtonian systems},
      journal = {J. Differential Equations},
      fjournal = {Journal of Differential Equations},
      volume = {82},
      year = {1989},
      number = {2},
      pages = {342--355},
      issn = {0022-0396},
      mrclass = {70F10 (58F10)},
      mrnumber = {1027973},
      mrreviewer = {Dorin Andrica},
      doi = {10.1016/0022-0396(89)90137-X},
      url = {https://doi.org/10.1016/0022-0396(89)90137-X},
      zblnumber = {0705.34034},
      }
  • [Shu] Go to document M. Shub, "Appendix to Smale’s paper: “Diagonals and relative equilibria”," in Manifolds — Amsterdam 1970, 1971, pp. 199-201.
    @INPROCEEDINGS{Shu,
      author = {Shub, M.},
      title = {Appendix to {S}male's paper: ``{D}iagonals and relative equilibria''},
      booktitle = {Manifolds -- {A}msterdam 1970},
      venue = {{P}roc. {N}uffic {S}ummer {S}chool},
      series = {Lecture Notes in Math.},
      volume = {197},
      pages = {199--201},
      publisher = {Springer, Berlin},
      year = {1971},
      mrclass = {85.57},
      mrnumber = {0278700},
      mrreviewer = {J. W. Robbin},
      doi = {10.1007/BFb0068619},
      url = {https://doi.org/10.1007/BFb0068619},
      zblnumber = {0219.57026},
      }
  • [Sit] Go to document K. Sitnikov, "The existence of oscillatory motions in the three-body problems," Soviet Physics. Dokl., vol. 5, pp. 647-650, 1960.
    @ARTICLE{Sit,
      author = {Sitnikov, K.},
      title = {The existence of oscillatory motions in the three-body problems},
      journal = {Soviet Physics. Dokl.},
      fjournal = {Soviet Physics. Doklady},
      volume = {5},
      year = {1960},
      pages = {647--650},
      issn = {0038-5689},
      mrclass = {85.34},
      mrnumber = {0127389},
      mrreviewer = {E. Leimanis},
      url = {http://mi.mathnet.ru/eng/dan/v133/i2/p303},
      zblnumber = {0108.18603},
      }
  • [Win] A. Wintner, The Analytical Foundations of Celestial Mechanics, Princeton Univ. Press, Princeton, N. J., 1941.
    @BOOK{Win,
      author = {Wintner, Aurel},
      title = {The {A}nalytical {F}oundations of {C}elestial {M}echanics},
      series = {Princeton Mathematical Series, v. 5},
      publisher = {Princeton Univ. Press, Princeton, N. J.},
      year = {1941},
      pages = {xii+448},
      mrclass = {85.0X},
      mrnumber = {0005824},
      mrreviewer = {E. J. Moulton},
      zblnumber = {0026.02302},
      }
  • [Xia] Go to document Z. Xia, "The existence of noncollision singularities in Newtonian systems," Ann. of Math. (2), vol. 135, iss. 3, pp. 411-468, 1992.
    @ARTICLE{Xia,
      author = {Xia, Zhihong},
      title = {The existence of noncollision singularities in {N}ewtonian systems},
      journal = {Ann. of Math. (2)},
      fjournal = {Annals of Mathematics. Second Series},
      volume = {135},
      year = {1992},
      number = {3},
      pages = {411--468},
      issn = {0003-486X},
      mrclass = {70F10 (58F05 70F15 70F35)},
      mrnumber = {1166640},
      mrreviewer = {Ernesto A. Lacomba},
      doi = {10.2307/2946572},
      url = {https://doi.org/10.2307/2946572},
      zblnumber = {0764.70006},
      }
  • [YuZha] Go to document X. Yu and S. Zhang, "Action-minimizing solutions of the one-dimensional $N$-body problem," Celestial Mech. Dynam. Astronom., vol. 130, iss. 5, p. 37, 2018.
    @ARTICLE{YuZha,
      author = {Yu, Xiang and Zhang, Shiqing},
      title = {Action-minimizing solutions of the one-dimensional {$N$}-body problem},
      journal = {Celestial Mech. Dynam. Astronom.},
      fjournal = {Celestial Mechanics \& Dynamical Astronomy. An International Journal of Space Dynamics},
      volume = {130},
      year = {2018},
      number = {5},
      pages = {Paper No. 37, 15},
      issn = {0923-2958},
      mrclass = {70F10 (34B15 70F16 70G75)},
      mrnumber = {3798029},
      mrreviewer = {Khalil Zare},
      doi = {10.1007/s10569-018-9830-z},
      url = {https://doi.org/10.1007/s10569-018-9830-z},
      zblnumber = {1391.70036},
      }
  • [Zei] H. relax von Zeipel, "Sur les singularités du probléme des $n$ corps," Ark. Math. Astr. Fys., iss. 4, pp. 1-4, 1908.
    @ARTICLE{Zei,
      author = {{\relax von Zeipel},
      H.},
      title = {Sur les singularités du probléme des $n$ corps},
      journal = {Ark. Math. Astr. Fys.},
      number = {4},
      year = {1908},
      pages = {1--4},
      zblnumber = {},
      }

Authors

Ezequiel Maderna

IMERL & CMAT, Universidad de la República, Montevideo, Uruguay

Andrea Venturelli

Laboratoire de Mathématiques d'Avignon, Avignon, France