Inventiones mathematicae

, Volume 185, Issue 2, pp 283–332 | Cite as

Platonic polyhedra, topological constraints and periodic solutions of the classical N-body problem



We prove the existence of a number of smooth periodic motions u of the classical Newtonian N-body problem which, up to a relabeling of the N particles, are invariant under the rotation group \(\mathcal{R}\) of one of the five Platonic polyhedra. The number N coincides with the order \(|\mathcal{R}|\) of \(\mathcal{R}\) and the particles have all the same mass. Our approach is variational and u is a minimizer of the Lagrangian action \(\mathcal{A}\) on a suitable subset \(\mathcal{K}\) of the H 1 T-periodic maps u:ℝ→ℝ3N . The set \({\mathcal {K}}\) is a cone and is determined by imposing on u both topological and symmetry constraints which are defined in terms of the rotation group \(\mathcal{R}\). There exist infinitely many such cones \({\mathcal {K}}\), all with the property that \({\mathcal {A}}|_{{\mathcal {K}}}\) is coercive. For a certain number of them, using level estimates and local deformations, we show that minimizers are free of collisions and therefore classical solutions of the N-body problem with a rich geometric–kinematic structure.


Periodic Solution Periodic Orbit Periodic Motion Rotation Group Periodic Sequence 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Albouy, A.: Lectures on the two-body problem. In: Cabral, H., Diacu, F. (eds.) Classical and Celestial Mechanics (The Recife Lectures). Princeton University Press, Princeton (2002) Google Scholar
  2. 2.
    Barutello, V., Ferrario, D.L., Terracini, S.: On the singularities of generalized solutions to n-body-type problems. Int. Math. Res. Not. 069, 1–78 (2008) Google Scholar
  3. 3.
    Chen, K.C.: Binary decomposition for planar N-body problems and symmetric periodic solutions. Arch. Ration. Mech. Anal. 170, 247–276 (2003) MATHCrossRefMathSciNetGoogle Scholar
  4. 4.
    Chenciner, A.: Action minimizing solutions of the Newtonian n-body problem: from homology to symmetry. In: ICM 2002, Beijing (2002) Google Scholar
  5. 5.
    Chenciner, A.: Symmetries and “simple” solutions of the classical N-body problem. In: ICMP03 (2003) Google Scholar
  6. 6.
    Chenciner, A.: Private communication Google Scholar
  7. 7.
    Chenciner, A., Montgomery, R.: A remarkable periodic solution of the three-body problem in the case of equal masses. Ann. Math. (2) 152, 881–901 (2000) MATHCrossRefMathSciNetGoogle Scholar
  8. 8.
    Chenciner, A., Venturelli, A.: Minima de l’intégrale d’action du Problème newtonien de 4 corps de masses égales dans ℝ3: orbites “hip–hop”. Celest. Mech. Dyn. Astron. 77, 139–152 (2000) MATHCrossRefMathSciNetGoogle Scholar
  9. 9.
    Chenciner, A., Gerver, J., Montgomenty, R., Simó, C.: Simple choreographic motions of N-bodies: a preliminary study. In: Geometry, Mechanics, and Dynamics, pp. 287–308. Springer, New York (2002) CrossRefGoogle Scholar
  10. 10.
    Coti Zelati, V.: Periodic solutions for N-body type problems. Ann. Inst. Henri Poincaré, Anal. Non Linéaire 7/5, 477–492 (1990) MathSciNetGoogle Scholar
  11. 11.
    Coxeter, H.S.M., Longuet-Higgins, M.S., Miller, J.C.P.: Uniform polyhedra. Philos. Trans. R. Soc. Lond. 246(916), 401–450 (1954) MATHCrossRefMathSciNetGoogle Scholar
  12. 12.
    Cundy, H., Rollett, A.P.: Mathematical Models. Clarendon, Oxford (1954) Google Scholar
  13. 13.
    Dacorogna, B.: Direct Methods in the Calculus of Variations. Applied Mathematical Sciences, vol. 78. Springer, Berlin (1989) MATHGoogle Scholar
  14. 14.
    Davies, M., Truman, A., Williams, D.: Classical periodic solutions of the equal-mass 2n-body problem, n-ion problem and the n-electron atom problem. Phys. Lett. A 99(1), 15–18 (1983) CrossRefMathSciNetGoogle Scholar
  15. 15.
    Degiovanni, M., Giannoni, F., Marino, A.: Dynamical systems with Newtonian type potentials. Ann. Sc. Norm. Super. Pisa, Cl. Sci. 15, 467–494 (1988) MATHGoogle Scholar
  16. 16.
    Delgado, J., Vidal, C.: The tetrahedral 4-body problem. J. Dyn. Differ. Equ. 11(4), 735–780 (1999) MATHCrossRefMathSciNetGoogle Scholar
  17. 17.
    Ferrario, D.L.: Transitive decomposition of symmetry groups for the n-body problem. Adv. Math. 213(2), 763–784 (2007) MATHCrossRefMathSciNetGoogle Scholar
  18. 18.
    Ferrario, D.L., Portaluri, A.: On the dihedral n-body problem. Nonlinearity 21, 1307–1321 (2008) MATHCrossRefMathSciNetGoogle Scholar
  19. 19.
    Ferrario, D., Terracini, S.: On the existence of collisionless equivariant minimizers for the classical n-body problem. Invent. Math. 155, 305–362 (2004) MATHCrossRefMathSciNetGoogle Scholar
  20. 20.
    Giusti, E.: Direct Methods in the Calculus of Variations. World Scientific, Singapore (2002) Google Scholar
  21. 21.
    Gordon, W.B.: Conservative dynamical systems involving strong forces. Trans. Am. Math. Soc. 204, 113–135 (1975) MATHCrossRefGoogle Scholar
  22. 22.
    Gordon, W.B.: A minimizing property of Keplerian orbits. Am. J. Math. 99(5), 961–971 (1977) MATHCrossRefGoogle Scholar
  23. 23.
    Grove, L.C., Benson, C.T.: Finite Reflection Groups. Springer, Berlin (1985) MATHGoogle Scholar
  24. 24.
  25. 25.
  26. 26.
  27. 27.
    Marchal, C.: How the method of minimization of action avoid singularities. Celest. Mech. Dyn. Astron. 83, 325–353 (2002) MATHCrossRefMathSciNetGoogle Scholar
  28. 28.
    Montgomery, R.: The n-body problem, the braid group and action minimizing periodic solutions. Nonlinearity 11, 363–376 (1998) MATHCrossRefMathSciNetGoogle Scholar
  29. 29.
    Moore, C.: Braids in classical dynamics. Phys. Rev. Lett. 70(24), 3675–3679 (1993) MATHCrossRefMathSciNetGoogle Scholar
  30. 30.
    Moore, C., Nauenberg, M.: New periodic orbits for the n-body problem. Preprint (2008). arXiv:math/0511219v1
  31. 31.
    Palais, R.: The principle of symmetric criticality. Commun. Math. Phys. 69(1), 19–30 (1979) MATHCrossRefMathSciNetGoogle Scholar
  32. 32.
    Poincaré, H.: Sur Les Solutions Périodiques et le Principe de Moindre Action. C. R. Acad. Sci. 123, 915–918 (1896) MATHGoogle Scholar
  33. 33.
    Saari, D.G.: Collisions, rings, and other newtonian N-body problems, In: CBMS, vol. 104. Am. Math. Soc., Providence (2005) Google Scholar
  34. 34.
    Simó, C.: New Families of Solutions in N-Body Problems. In: Casacuberta et al. (eds.) Proceedings of the Third European Congress of Mathematics. Progress in Mathematics, vol. 201, pp. 101–115 (2001) Google Scholar
  35. 35.
    Terracini, S.: On the variational approach to the periodic n-body problem. Celest. Mech. Dyn. Astron. 95(1–4), 3–25 (2006) MATHCrossRefMathSciNetGoogle Scholar
  36. 36.
    Terracini, S., Venturelli, A.: Symmetric trajectories for the 2N-body problem with equal masses. Arch. Ration. Mech. Anal. 184, 465–493 (2007) MATHCrossRefMathSciNetGoogle Scholar
  37. 37.
    Venturelli, A.: Une caracterisation variationelle des solutions de Lagrange du problème plan des trois corps. C. R. Math. 332(I), 641–644 (2001) MATHMathSciNetGoogle Scholar
  38. 38.
    Venturelli, A.: Application de la minimisation de l’action au Problème de N corps dans le plan e dans l’espace. Thesis, University Paris VII 2002 Google Scholar
  39. 39.
    Vidal, C.: The tetrahedral 4-body problem with rotation. Celest. Mech. Dyn. Astron. 71(1), 15–33 (1998/1999) CrossRefMathSciNetGoogle Scholar
  40. 40.
    Wintner, A.: The Analytical Foundations of Celestial Mechanics. Princeton University Press, Princeton (1941) Google Scholar

Copyright information

© Springer-Verlag 2010

Authors and Affiliations

  1. 1.Dipartimento di Matematica Pura ed ApplicataUniversità di L’AquilaL’AquilaItaly
  2. 2.Dipartimento di MatematicaUniversità di PisaPisaItaly
  3. 3.Dipartimento di MatematicaUniversità di Roma ‘La Sapienza’RomeItaly

Personalised recommendations