Singular dynamics under a weak potential on a sphere

详细信息    查看全文

• 作者：Roberto Castelli (1)
  Francesco Paparella (2)
  Alessandro Portaluri (2)
  
• 关键词：Primary 70F10 ; Secondary 37C80 ; Singular dynamics ; McGehee coordinates ; Regularization of collisions ; Heteroclinics
• 刊名：NoDEA : Nonlinear Differential Equations and Applications
• 出版年：2013
• 出版时间：June 2013
• 年：2013
• 卷：20
• 期：3
• 页码：845-872
• 全文大小：885KB
• 参考文献：1. Ralph, A., Marsden, J.E.: Foundations of Mechanics, 2nd edn. Benjamin/Cummings, Ink. Massachusetts (1978)
  2. Vivina, B., Ferrario, D.L., Susanna, T.: On the singularities of generalized solutions to / n-body-type problems. Int. Math. Res. Not. IMRN (2008)
  3. Castelli, R.: On the variational approach to the one and N-centre problem with weak forces. Ph.D. dissertation, University of Milano-Bicocca (2009)
  4. Roberto C., Susanna T.: On the regularization of the collision solutions of the one-center problem with weak forces. Discrete Contin. Dyn. Syst 31(4), 1197鈥?218 (2011)
  5. Diacu F., Perez-Chavela E., Reyes Victoria J.G.: An intrinsic approach in the curved n-body problem: the negative curvature case. J. Differ. Equ. 252, 4529鈥?562 (2012) CrossRef
  6. Florin D.: Polygonal homographic orbits of the curved / n-body problem. Trans. Am. Math. Soc. 364(5), 2783鈥?802 (2012) CrossRef
  7. Florin D., Perez-Chavela E.: Homographic solutions of the curved 3-body problem. J. Differ. Equ. 250, 1747鈥?766 (2011) CrossRef
  8. Florin, D.: On the singularities of the curved / n-body problem. Trans. Am. Math. Soc. 363(4), 2249鈥?264 (2011). http://arxiv.org/abs/0812.3333
  9. Florin, D., Perez-Chavela, E., Santoprete, M.: The / n-body problem in spaces of constant curvature. http://arxiv.org/abs/0807.1747
  10. Easton R.: Regularization of vector fields by surgery. J. Differ. Equ. 10, 92鈥?9 (1971) CrossRef
  11. Ferrario D.L., Portaluri A.: On the dihedral / n-body problem. Nonlinearity 21(6), 1307鈥?321 (2008) CrossRef
  12. Bellettini G., Fusco G., Gronchi G.F.: Regularization of the two-body problem via smoothing the potential. Commun. Pure Appl. Anal. 2(3), 323鈥?53 (2003) CrossRef
  13. Hooverman R.H.: Charged particle orbits in a logarithmic potential. J. Appl. Phys 34, 3505鈥?508 (1963) CrossRef
  14. Kenig C.E., Ponce G., Vega L.: On the interaction of nearly parallel vortex filaments. Commun. Math. Phys. 243, 471鈥?83 (2003) CrossRef
  15. Levi-Civita T.: Sur la r茅gularisation du probl猫me des trois corps. Acta Math. 42(1), 99鈥?44 (1920) CrossRef
  16. McGehee R.: Triple collision in the collinear three-body problem. Invent. Math. 27, 191鈥?27 (1974) CrossRef
  17. Newton, P.: The / N-vortex problem. Analytical techniques. Applied Mathematical Sciences, vol. 145. Springer, New York (2001)
  18. Stoica C., Font A.: Global dynamics in the singular logarithmic potential. J. Phys. A 36, 7693鈥?714 (2003) CrossRef
  19. Touma J., Tremaine S.: A map for eccentric orbits in non-axisymmetric potentials. MNRAS 292, 905鈥?32 (1997) CrossRef
  20. Vidal C.: The tetrahedral 4-body problem with rotation. Celestial Mech. Dyn. Astron. 71(1), 15鈥?3 (1999) CrossRef
  
• 作者单位：Roberto Castelli (1)
  Francesco Paparella (2)
  Alessandro Portaluri (2)
  
  1. Basque Center for Applied Mathematics (BCAM), Alameda de Mazarredo, 14, 48009, Bilbao, Spain
  2. Dipartimento di Matematica e Fisica 鈥淓nnio De Giorgi鈥? Ex-collegio Fiorini, University of Salento, 73100, Lecce, Italy
  
• ISSN：1420-9004

文摘

We give a detailed analytical description of the global dynamics of a point mass moving on a sphere under the action of a logarithmic potential. We perform a McGehee-type blow-up in order to cope with the singularity of the potential when the point mass goes through the singularity. In addition we investigate the rest-points of the flow, the invariant (stable and unstable) manifolds and we give a complete dynamical description of the motion.