American Institute of Mathematical Sciences

Advanced
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

Francesco Paparella 1, and  Alessandro Portaluri 2,

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.
Keywords: logarithmic potential, global dynamics, McGehee coordinates, n-body problems., Dihedral $N$-vortex filaments.
Mathematics Subject Classification: Primary: 70F10; Secondary: 37C80, 76B4.
Citation: Francesco Paparella, Alessandro Portaluri. Geometry of stationary solutions for a system of vortex filaments: A dynamical approach. Discrete & Continuous Dynamical Systems - A, 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, (2008).   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.  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.  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.  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.  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.  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.  doi: 10.1007/BF01390017.  Google Scholar

[10]

R. L. Devaney, Singularities in classical mechanical systems,, in, 10 (1981), 1979.   Google Scholar

[11]

F. Diacu, Regularization of partial collisions in the N -body problem,, Differential Integral Equations, 5 (1992), 103.   Google Scholar

[12]

Davide L. Ferrario, Transitive decomposition of symmetry groups for the $n$-body problem,, Adv. in Math., 213 (2007), 763.  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.  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.  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 (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.  doi: 10.1017/S0022112095001121.  Google Scholar

[18]

R. McGehee, Triple collision in the collinear three-body problem,, Invent. Math., 27 (1974), 191.   Google Scholar

[19]

Paul Newton, "The $N$-Vortex Problem. Analytical Techniques,", Applied Mathematical Sciences, 145 (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.  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.   Google Scholar

[22]

H. Pollard and D. G. Saari, Singularities of the n-body problem. II. In Inequalities, II,, (Proc. Second Sympos., (1970), 255.   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.   Google Scholar

[26]

K. F. Sundman, Nouvelles recherches sur le probleme des trois corps,, Acta Soc. Sci. Fenn., 35 (1909).   Google Scholar

[27]

Cristina Stoica and Andreea Font, Global dynamics in the singular logarithmic potential,, J. Phys. A, 36 (2003), 7693.  doi: 10.1088/0305-4470/36/28/302.  Google Scholar

[28]

A. Wintner, "The Analytical Foundations of Celestial Mechanics,", Princeton Mathematical Series, 5 (1941).   Google Scholar

show all references

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, (2008).   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.  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.  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.  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.  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.  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.  doi: 10.1007/BF01390017.  Google Scholar

[10]

R. L. Devaney, Singularities in classical mechanical systems,, in, 10 (1981), 1979.   Google Scholar

[11]

F. Diacu, Regularization of partial collisions in the N -body problem,, Differential Integral Equations, 5 (1992), 103.   Google Scholar

[12]

Davide L. Ferrario, Transitive decomposition of symmetry groups for the $n$-body problem,, Adv. in Math., 213 (2007), 763.  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.  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.  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 (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.  doi: 10.1017/S0022112095001121.  Google Scholar

[18]

R. McGehee, Triple collision in the collinear three-body problem,, Invent. Math., 27 (1974), 191.   Google Scholar

[19]

Paul Newton, "The $N$-Vortex Problem. Analytical Techniques,", Applied Mathematical Sciences, 145 (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.  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.   Google Scholar

[22]

H. Pollard and D. G. Saari, Singularities of the n-body problem. II. In Inequalities, II,, (Proc. Second Sympos., (1970), 255.   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.   Google Scholar

[26]

K. F. Sundman, Nouvelles recherches sur le probleme des trois corps,, Acta Soc. Sci. Fenn., 35 (1909).   Google Scholar

[27]

Cristina Stoica and Andreea Font, Global dynamics in the singular logarithmic potential,, J. Phys. A, 36 (2003), 7693.  doi: 10.1088/0305-4470/36/28/302.  Google Scholar

[28]

A. Wintner, "The Analytical Foundations of Celestial Mechanics,", Princeton Mathematical Series, 5 (1941).   Google Scholar

[1]

Chjan C. Lim, Joseph Nebus, Syed M. Assad. Monte-Carlo and polyhedron-based simulations I: extremal states of the logarithmic N-body problem on a sphere. Discrete & Continuous Dynamical Systems - B, 2003, 3 (3) : 313-342. doi: 10.3934/dcdsb.2003.3.313
[2]

Luis Vega. The dynamics of vortex filaments with corners. Communications on Pure & Applied Analysis, 2015, 14 (4) : 1581-1601. doi: 10.3934/cpaa.2015.14.1581
[3]

Shiqing Zhang, Qing Zhou. Nonplanar and noncollision periodic solutions for $N$-body problems. Discrete & Continuous Dynamical Systems - A, 2004, 10 (3) : 679-685. doi: 10.3934/dcds.2004.10.679
[4]

Davide L. Ferrario, Alessandro Portaluri. Dynamics of the the dihedral four-body problem. Discrete & Continuous Dynamical Systems - S, 2013, 6 (4) : 925-974. doi: 10.3934/dcdss.2013.6.925
[5]

Younghun Hong. Strichartz estimates for $N$-body Schrödinger operators with small potential interactions. Discrete & Continuous Dynamical Systems - A, 2017, 37 (10) : 5355-5365. doi: 10.3934/dcds.2017233
[6]

P.K. Newton. N-vortex equilibrium theory. Discrete & Continuous Dynamical Systems - A, 2007, 19 (2) : 411-418. doi: 10.3934/dcds.2007.19.411
[7]

Nai-Chia Chen. Symmetric periodic orbits in three sub-problems of the $N$-body problem. Discrete & Continuous Dynamical Systems - B, 2014, 19 (6) : 1523-1548. doi: 10.3934/dcdsb.2014.19.1523
[8]

Juan J. Morales-Ruiz, Sergi Simon. On the meromorphic non-integrability of some $N$-body problems. Discrete & Continuous Dynamical Systems - A, 2009, 24 (4) : 1225-1273. doi: 10.3934/dcds.2009.24.1225
[9]

Sun-Sig Byun, Yumi Cho, Shuang Liang. Calderón-Zygmund estimates for quasilinear elliptic double obstacle problems with variable exponent and logarithmic growth. Discrete & Continuous Dynamical Systems - B, 2017, 22 (11) : 0-0. doi: 10.3934/dcdsb.2020038
[10]

Gabriella Pinzari. Global Kolmogorov tori in the planetary $\boldsymbol N$-body problem. Announcement of result. Electronic Research Announcements, 2015, 22: 55-75. doi: 10.3934/era.2015.22.55
[11]

P.K. Newton, M. Ruith, E. Upchurch. The constrained planar N-vortex problem: I. Integrability. Discrete & Continuous Dynamical Systems - B, 2005, 5 (1) : 137-152. doi: 10.3934/dcdsb.2005.5.137
[12]

Björn Gebhard. Periodic solutions for the N-vortex problem via a superposition principle. Discrete & Continuous Dynamical Systems - A, 2018, 38 (11) : 5443-5460. doi: 10.3934/dcds.2018240
[13]

Chiun-Chuan Chen, Li-Chang Hung, Chen-Chih Lai. An N-barrier maximum principle for autonomous systems of $n$ species and its application to problems arising from population dynamics. Communications on Pure & Applied Analysis, 2019, 18 (1) : 33-50. doi: 10.3934/cpaa.2019003
[14]

Eduardo S. G. Leandro. On the Dziobek configurations of the restricted $(N+1)$-body problem with equal masses. Discrete & Continuous Dynamical Systems - S, 2008, 1 (4) : 589-595. doi: 10.3934/dcdss.2008.1.589
[15]

Marshall Hampton, Anders Nedergaard Jensen. Finiteness of relative equilibria in the planar generalized $N$-body problem with fixed subconfigurations. Journal of Geometric Mechanics, 2015, 7 (1) : 35-42. doi: 10.3934/jgm.2015.7.35
[16]

Montserrat Corbera, Jaume Llibre. On the existence of bi--pyramidal central configurations of the $n+2$--body problem with an $n$--gon base. Discrete & Continuous Dynamical Systems - A, 2013, 33 (3) : 1049-1060. doi: 10.3934/dcds.2013.33.1049
[17]

Nicola Soave, Susanna Terracini. Symbolic dynamics for the $N$-centre problem at negative energies. Discrete & Continuous Dynamical Systems - A, 2012, 32 (9) : 3245-3301. doi: 10.3934/dcds.2012.32.3245
[18]

Fernando Jiménez, Jürgen Scheurle. On the discretization of nonholonomic dynamics in $\mathbb{R}^n$. Journal of Geometric Mechanics, 2015, 7 (1) : 43-80. doi: 10.3934/jgm.2015.7.43
[19]

Perikles G. Papadopoulos, Nikolaos M. Stavrakakis. Global existence for a wave equation on $R^n$. Discrete & Continuous Dynamical Systems - S, 2008, 1 (1) : 139-149. doi: 10.3934/dcdss.2008.1.139
[20]

Sergey V. Bolotin, Piero Negrini. Global regularization for the $n$-center problem on a manifold. Discrete & Continuous Dynamical Systems - A, 2002, 8 (4) : 873-892. doi: 10.3934/dcds.2002.8.873

2018 Impact Factor: 1.143

Tools

Metrics

  • PDF downloads (28)
  • HTML views (0)
  • Cited by (1)

Other articles
by authors

[Back to Top]

Copyright © 2020 American Institute of Mathematical Sciences