# American Institute of Mathematical Sciences

July  2013, 33(7): 3011-3042. doi: 10.3934/dcds.2013.33.3011

## Geometry of stationary solutions for a system of vortex filaments: A dynamical approach

 1 Dipartimento di Matematica e Fisica "Ennio De Giorgi", Università del Salento, 73100, Lecce 2 Dipartimento di Matematica e Fisica “Ennio De Giorgi”, Università del Salento, 73100 Lecce

Received  April 2012 Revised  October 2012 Published  January 2013

We give a detailed analytical description of the global dynamics of $N$ points interacting through the singular logarithmic potential and subject to the following symmetry constraint: at each instant they form an orbit of the dihedral group $D_l$ of order $2l$. The main device in order to achieve our results is a technique very popular in Celestial Mechanics, usually referred to as McGehee transformation. After performing this change of coordinates that regularizes the total collision, we study the rest-points of the flow, the invariant manifolds and, with the help of a computer algebra system, we derive interesting information about the global dynamics for $l=2$. We observe that our problem is equivalent to studying the geometry of stationary configurations of nearly-parallel vortex filaments in three dimensions in the LIA approximation.
Citation: Francesco Paparella, Alessandro Portaluri. Geometry of stationary solutions for a system of vortex filaments: A dynamical approach. Discrete & Continuous Dynamical Systems, 2013, 33 (7) : 3011-3042. doi: 10.3934/dcds.2013.33.3011
##### References:
 [1] Vivina Barutello, Davide L. Ferrario and Susanna Terracini, On the singularities of generalized solutions to $n$-body-type problems, Int. Math. Res. Not. IMRN 2008, Art. ID rnn 069, 78 pp.  Google Scholar [2] G. Bellettini, G. Fusco and G. F. Gronchi, Regularization of the two body problem via smoothing the potential, Commun. Pure Appl. Anal., 2 (2003), 323-353. doi: 10.3934/cpaa.2003.2.323.  Google Scholar [3] Anna Capietto, Francesca Dalbono and Alessandro Portaluri, A multiplicity result for a class of strongly indefinite asymptotically linear second order systems, Nonlinear Anal., 72 (2010), 2874-2890. doi: 10.1016/j.na.2009.11.032.  Google Scholar [4] Castelli Roberto, "Moti Periodici di Filamenti Vorticosi Quasi-Paralleli," Laurea Magistrale dissertation at University of Milano-Bicocca, 2004. Google Scholar [5] R. Castelli, F. Paparella and A. Portaluri, Singular dynamics under a weak potential on a sphere,, To appear in NoDEA. , ().  doi: 10.1007/s00030-012-0182-1.  Google Scholar [6] Castelli Roberto and Terracini Susanna, On the regularization of the collision solutions of the one-center problem with weak forces, Discrete Contin. Dyn. Syst., 31 (2011), 1197-1218. doi: 10.3934/dcds.2011.31.1197.  Google Scholar [7] F. Dalbono and A. Portaluri, Morse-Smale index theorems for elliptic boundary deformation problems, Journal of Differential Equations, 253 (2012), 463-480. doi: 10.1016/j.jde.2012.04.008.  Google Scholar [8] Ennio De Giorgi, Conjectures concerning some evolution problems, Duke Math. J., 81 (1996), 255-268. doi: 10.1215/S0012-7094-96-08114-4.  Google Scholar [9] R. L. Devaney, Triple collision in the planar isosceles three-body problem, Invent. Math., 60 (1980), 249-267. doi: 10.1007/BF01390017.  Google Scholar [10] R. L. Devaney, Singularities in classical mechanical systems, in "Ergodic Theory and Dynamical Systems, I" (College Park, Md., 1979-80), 10 of Progr. Math. Birkhäuser Boston, Mass., (1981), 211-333.  Google Scholar [11] F. Diacu, Regularization of partial collisions in the N -body problem, Differential Integral Equations, 5 (1992), 103-136.  Google Scholar [12] Davide L. Ferrario, Transitive decomposition of symmetry groups for the $n$-body problem, Adv. in Math., 213 (2007), 763-784. doi: 10.1016/j.aim.2007.01.009.  Google Scholar [13] Davide L. Ferrario and Alessandro Portaluri, On the dihedral $n$- body problem, Nonlinearity, 21 (2008), 1307-1321. doi: 10.1088/0951-7715/21/6/009.  Google Scholar [14] Davide L. Ferrario and Alessandro Portaluri, Dynamics of the the dihedral four-body problem,, To appear in DCDS-S, ().   Google Scholar [15] Roberto Giambò, Paolo Piccione and Alessandro Portaluri, Computation of the Maslov index and the spectral flow via partial signatures, C. R. Math. Acad. Sci. Paris, 338 (2004), 397-402. doi: 10.1016/j.crma.2004.01.004.  Google Scholar [16] M. W. Hirsch, C. C. Pugh and M. Shub, "Invariant Manifolds," Lecture Notes in Mathematics, 583. Springer-Verlag, Berlin-New York, 1977.  Google Scholar [17] Rupert Klein, Andrew J. Majda and Kumaran Damodaran, Simplified equations for the interaction of nearly parallel vortex filaments, J. Fluid Mech., 288 (1995), 201-248. doi: 10.1017/S0022112095001121.  Google Scholar [18] R. McGehee, Triple collision in the collinear three-body problem, Invent. Math., 27 (1974), 191-227.  Google Scholar [19] Paul Newton, "The $N$-Vortex Problem. Analytical Techniques," Applied Mathematical Sciences, 145. Springer-Verlag, New York, 2001. doi: 10.1007/978-1-4684-9290-3.  Google Scholar [20] F. Paparella and A. Portaluri, Dynamics of (4 + 1)-Dihedrally symmetric nearly parallel vortex filaments, Acta. Appl. Math., 122 (2012), 349-366. doi: 10.1007/s10440-012-9748-5.  Google Scholar [21] H. Pollard and D. G. Saari, Singularities of the n-body problem. I, Arch. Rational Mech. Anal., 30 (1968), 263-269.  Google Scholar [22] H. Pollard and D. G. Saari, Singularities of the n-body problem. II. In Inequalities, II, (Proc. Second Sympos., U.S. Air Force Acad., Colo., 1967), 255-259. Academic Press, New York, (1970).  Google Scholar [23] Alessandro Portaluri, Maslov index for Hamiltonian systems,, Electron. J. Differential Equations, 2008 ().   Google Scholar [24] D. G. Saari, Singularities and collisions of Newtonian gravitational systems,, Arch. Rational Mech. Anal., 49 (): 311.   Google Scholar [25] H. J. Sperling, On the real singularities of the $N$ -body problem, J. Reine Angew. Math., 245 (1970), 15-40.  Google Scholar [26] K. F. Sundman, Nouvelles recherches sur le probleme des trois corps, Acta Soc. Sci. Fenn., 35 (1909), 9. Google Scholar [27] Cristina Stoica and Andreea Font, Global dynamics in the singular logarithmic potential, J. Phys. A, 36 (2003), 7693-7714. doi: 10.1088/0305-4470/36/28/302.  Google Scholar [28] A. Wintner, "The Analytical Foundations of Celestial Mechanics," Princeton Mathematical Series, 5, Princeton University Press, Princeton, N. J., 1941.  Google Scholar

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##### References:
 [1] Vivina Barutello, Davide L. Ferrario and Susanna Terracini, On the singularities of generalized solutions to $n$-body-type problems, Int. Math. Res. Not. IMRN 2008, Art. ID rnn 069, 78 pp.  Google Scholar [2] G. Bellettini, G. Fusco and G. F. Gronchi, Regularization of the two body problem via smoothing the potential, Commun. Pure Appl. Anal., 2 (2003), 323-353. doi: 10.3934/cpaa.2003.2.323.  Google Scholar [3] Anna Capietto, Francesca Dalbono and Alessandro Portaluri, A multiplicity result for a class of strongly indefinite asymptotically linear second order systems, Nonlinear Anal., 72 (2010), 2874-2890. doi: 10.1016/j.na.2009.11.032.  Google Scholar [4] Castelli Roberto, "Moti Periodici di Filamenti Vorticosi Quasi-Paralleli," Laurea Magistrale dissertation at University of Milano-Bicocca, 2004. Google Scholar [5] R. Castelli, F. Paparella and A. Portaluri, Singular dynamics under a weak potential on a sphere,, To appear in NoDEA. , ().  doi: 10.1007/s00030-012-0182-1.  Google Scholar [6] Castelli Roberto and Terracini Susanna, On the regularization of the collision solutions of the one-center problem with weak forces, Discrete Contin. Dyn. Syst., 31 (2011), 1197-1218. doi: 10.3934/dcds.2011.31.1197.  Google Scholar [7] F. Dalbono and A. Portaluri, Morse-Smale index theorems for elliptic boundary deformation problems, Journal of Differential Equations, 253 (2012), 463-480. doi: 10.1016/j.jde.2012.04.008.  Google Scholar [8] Ennio De Giorgi, Conjectures concerning some evolution problems, Duke Math. J., 81 (1996), 255-268. doi: 10.1215/S0012-7094-96-08114-4.  Google Scholar [9] R. L. Devaney, Triple collision in the planar isosceles three-body problem, Invent. Math., 60 (1980), 249-267. doi: 10.1007/BF01390017.  Google Scholar [10] R. L. Devaney, Singularities in classical mechanical systems, in "Ergodic Theory and Dynamical Systems, I" (College Park, Md., 1979-80), 10 of Progr. Math. Birkhäuser Boston, Mass., (1981), 211-333.  Google Scholar [11] F. Diacu, Regularization of partial collisions in the N -body problem, Differential Integral Equations, 5 (1992), 103-136.  Google Scholar [12] Davide L. Ferrario, Transitive decomposition of symmetry groups for the $n$-body problem, Adv. in Math., 213 (2007), 763-784. doi: 10.1016/j.aim.2007.01.009.  Google Scholar [13] Davide L. Ferrario and Alessandro Portaluri, On the dihedral $n$- body problem, Nonlinearity, 21 (2008), 1307-1321. doi: 10.1088/0951-7715/21/6/009.  Google Scholar [14] Davide L. Ferrario and Alessandro Portaluri, Dynamics of the the dihedral four-body problem,, To appear in DCDS-S, ().   Google Scholar [15] Roberto Giambò, Paolo Piccione and Alessandro Portaluri, Computation of the Maslov index and the spectral flow via partial signatures, C. R. Math. Acad. Sci. Paris, 338 (2004), 397-402. doi: 10.1016/j.crma.2004.01.004.  Google Scholar [16] M. W. Hirsch, C. C. Pugh and M. Shub, "Invariant Manifolds," Lecture Notes in Mathematics, 583. Springer-Verlag, Berlin-New York, 1977.  Google Scholar [17] Rupert Klein, Andrew J. Majda and Kumaran Damodaran, Simplified equations for the interaction of nearly parallel vortex filaments, J. Fluid Mech., 288 (1995), 201-248. doi: 10.1017/S0022112095001121.  Google Scholar [18] R. McGehee, Triple collision in the collinear three-body problem, Invent. Math., 27 (1974), 191-227.  Google Scholar [19] Paul Newton, "The $N$-Vortex Problem. Analytical Techniques," Applied Mathematical Sciences, 145. Springer-Verlag, New York, 2001. doi: 10.1007/978-1-4684-9290-3.  Google Scholar [20] F. Paparella and A. Portaluri, Dynamics of (4 + 1)-Dihedrally symmetric nearly parallel vortex filaments, Acta. Appl. Math., 122 (2012), 349-366. doi: 10.1007/s10440-012-9748-5.  Google Scholar [21] H. Pollard and D. G. Saari, Singularities of the n-body problem. I, Arch. Rational Mech. Anal., 30 (1968), 263-269.  Google Scholar [22] H. Pollard and D. G. Saari, Singularities of the n-body problem. II. In Inequalities, II, (Proc. Second Sympos., U.S. Air Force Acad., Colo., 1967), 255-259. Academic Press, New York, (1970).  Google Scholar [23] Alessandro Portaluri, Maslov index for Hamiltonian systems,, Electron. J. Differential Equations, 2008 ().   Google Scholar [24] D. G. Saari, Singularities and collisions of Newtonian gravitational systems,, Arch. Rational Mech. Anal., 49 (): 311.   Google Scholar [25] H. J. Sperling, On the real singularities of the $N$ -body problem, J. Reine Angew. Math., 245 (1970), 15-40.  Google Scholar [26] K. F. Sundman, Nouvelles recherches sur le probleme des trois corps, Acta Soc. Sci. Fenn., 35 (1909), 9. Google Scholar [27] Cristina Stoica and Andreea Font, Global dynamics in the singular logarithmic potential, J. Phys. A, 36 (2003), 7693-7714. doi: 10.1088/0305-4470/36/28/302.  Google Scholar [28] A. Wintner, "The Analytical Foundations of Celestial Mechanics," Princeton Mathematical Series, 5, Princeton University Press, Princeton, N. J., 1941.  Google Scholar
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