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March  2015, 7(1): 35-42. doi: 10.3934/jgm.2015.7.35

Finiteness of relative equilibria in the planar generalized $N$-body problem with fixed subconfigurations

Marshall Hampton 1, and  Anders Nedergaard Jensen 2,

1. 

140 Solon Campus Center, 1117 University Dr., UMD, Duluth, MN 55812-3000, United States
2. 

Department of Mathematics, Ny Munkegade 118, DK-8000, Aarhus, Denmark

Received  December 2013 Revised  January 2015 Published  March 2015

We prove that a fixed configuration of $N-1$ masses in the plane can be extended to a central configuration of $N$ masses by adding a specified additional mass only in finitely many ways. This holds for a family of potential functions including the Newtonian gravitational case and the classical planar point vortex model.
Keywords: celestial mechanics, central configurations., vortices, relative equilibria, N-body problem.
Mathematics Subject Classification: Primary: 70F15, 70K42; Secondary: 14M2.
Citation: 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
References:
[1]

A. Albouy and A. Chenciner, Le problème des n corps et les distances mutuelles, Inv. Math., 131 (1998), 151-184. doi: 10.1007/s002220050200.  Google Scholar

[2]

A. Albouy and V. Kaloshin, Finiteness of central configurations of five bodies in the plane, Annals of Math., 176 (2012), 535-588. doi: 10.4007/annals.2012.176.1.10.  Google Scholar

[3]

R. Bieri and J. R. J. Groves, The geometry of the set of characters induced by valuations, J. Reine Angew. Math., 347 (1984), 168-195.  Google Scholar

[4]

L. M. Blumenthal and B. E. Gillam, Distribution of points in $n$-space, Amer. Math. Mon., 50 (1943), 181-185. doi: 10.2307/2302400.  Google Scholar

[5]

J. Chazy, Sur certaines trajectoires du problème des n corps, Bull. Astron., 35 (1918), 321-389. Google Scholar

[6]

L. Euler, De motu rectilineo trium corporum se mutuo attrahentium, Novi Comm. Acad. Sci. Imp. Petrop., 11 (1767), 144-151. Google Scholar

[7]

M. Hampton, Finiteness of kite relative equilibria in the five-vortex and five-body problems, Qual. Theory Dyn. Sys., 8 (2010), 349-356. doi: 10.1007/s12346-010-0016-7.  Google Scholar

[8]

M. Hampton and A. N. Jensen, Finiteness of spatial central configurations in the five-body problem, Cel. Mech. Dynam. Astron., 109 (2011), 321-332. doi: 10.1007/s10569-010-9328-9.  Google Scholar

[9]

M. Hampton and R. Moeckel, Finiteness of relative equilibria of the four-body problem, Inventiones Mathematicae, 163 (2006), 289-312. doi: 10.1007/s00222-005-0461-0.  Google Scholar

[10]

M. Hampton and R. Moeckel, Finiteness of stationary configurations of the four-vortex problem, Trans. Amer. Math. Soc., 361 (2009), 1317-1332. doi: 10.1090/S0002-9947-08-04685-0.  Google Scholar

[11]

H. Helmholtz, Uber Integrale der hydrodynamischen Gleichungen, Welche den Wirbelbewegungen entsprechen, Crelle's Journal für Mathematik, 55 (1858), 25-55; English translation by P. G. Tait, On the integrals of the hydrodynamical equations which express vortex-motion, Philosophical Magazine, (1867), 485-551. Google Scholar

[12]

A. N. Jensen, Algorithmic Aspects of Gröbner Fans and Tropical Varieties, Ph.D. Thesis, Department of Mathematical Sciences, University of Aarhus, Denmark, 2007. Google Scholar

[13]

A. N. Jensen, Gfan, a software system for Gröbner fans and tropical varieties,, , ().   Google Scholar

[14]

J. Kulevich, G. E. Roberts and C. Smith, Finiteness in the planar restricted four-body problem, Qual. Theory Dyn. Sys., 8 (2009), 357-370. doi: 10.1007/s12346-010-0006-9.  Google Scholar

[15]

J. L. Lagrange, Essai Sur le Problème des Trois Corps, Œuvres, 1772. Google Scholar

[16]

P. W. Lindstrom, The number of planar central configurations is finite when $N-1$ mass positions are fixed, Trans. Amer. Math. Soc., 353 (2001), 291-311. doi: 10.1090/S0002-9947-00-02568-X.  Google Scholar

[17]

D. Maglagan and B. Sturmfels, Introduction to Tropical Geometry, Expected publication date May 2015. Google Scholar

[18]

R. Moeckel, On central configurations, Math. Z., 205 (1990), 499-517. doi: 10.1007/BF02571259.  Google Scholar

[19]

R. Moeckel, Relative equilibria with clusters of small masses, J. Dyn. Diff. Eq., 9 (1997), 507-533. doi: 10.1007/BF02219396.  Google Scholar

[20]

R. Moeckel, Generic finiteness for Dziobek configurations, Trans. Amer. Math. Soc., 353 (2001), 4673-4686. doi: 10.1090/S0002-9947-01-02828-8.  Google Scholar

[21]

F. R. Moulton, The straight line solutions of the problem of n bodies, Ann. of Math., 12 (1910), 1-17. doi: 10.2307/2007159.  Google Scholar

[22]

I. Newton, Philosophi Naturalis Principia Mathematica, Royal Society, London, 1687. Google Scholar

[23]

G. Roberts, A continuum of relative equilibria in the five-body problem, Phys. D, 127 (1999), 141-145. doi: 10.1016/S0167-2789(98)00315-7.  Google Scholar

[24]

S. Smale, Mathematical problems for the next century, Mathematical Intelligencer, 20 (1998), 7-15. doi: 10.1007/BF03025291.  Google Scholar

[25]

D. Speyer and B. Sturmfels, The tropical Grassmannian, Adv. Geom., 4 (2004), 389-411. doi: 10.1515/advg.2004.023.  Google Scholar

[26]

F. Tien, Recursion Formulas of Central Configurations, Thesis, University Of Minnesota, 1993.  Google Scholar

[27]

A. Wintner, The Analytical Foundations of Celestial Mechanics, Princeton Math. Series, 5, Princeton University Press, Princeton, NJ, 1941.  Google Scholar

[28]

Z. Xia, Central configurations with many small masses, J. Differential Equations, 91 (1991), 168-179. doi: 10.1016/0022-0396(91)90137-X.  Google Scholar

show all references

References:
[1]

A. Albouy and A. Chenciner, Le problème des n corps et les distances mutuelles, Inv. Math., 131 (1998), 151-184. doi: 10.1007/s002220050200.  Google Scholar

[2]

A. Albouy and V. Kaloshin, Finiteness of central configurations of five bodies in the plane, Annals of Math., 176 (2012), 535-588. doi: 10.4007/annals.2012.176.1.10.  Google Scholar

[3]

R. Bieri and J. R. J. Groves, The geometry of the set of characters induced by valuations, J. Reine Angew. Math., 347 (1984), 168-195.  Google Scholar

[4]

L. M. Blumenthal and B. E. Gillam, Distribution of points in $n$-space, Amer. Math. Mon., 50 (1943), 181-185. doi: 10.2307/2302400.  Google Scholar

[5]

J. Chazy, Sur certaines trajectoires du problème des n corps, Bull. Astron., 35 (1918), 321-389. Google Scholar

[6]

L. Euler, De motu rectilineo trium corporum se mutuo attrahentium, Novi Comm. Acad. Sci. Imp. Petrop., 11 (1767), 144-151. Google Scholar

[7]

M. Hampton, Finiteness of kite relative equilibria in the five-vortex and five-body problems, Qual. Theory Dyn. Sys., 8 (2010), 349-356. doi: 10.1007/s12346-010-0016-7.  Google Scholar

[8]

M. Hampton and A. N. Jensen, Finiteness of spatial central configurations in the five-body problem, Cel. Mech. Dynam. Astron., 109 (2011), 321-332. doi: 10.1007/s10569-010-9328-9.  Google Scholar

[9]

M. Hampton and R. Moeckel, Finiteness of relative equilibria of the four-body problem, Inventiones Mathematicae, 163 (2006), 289-312. doi: 10.1007/s00222-005-0461-0.  Google Scholar

[10]

M. Hampton and R. Moeckel, Finiteness of stationary configurations of the four-vortex problem, Trans. Amer. Math. Soc., 361 (2009), 1317-1332. doi: 10.1090/S0002-9947-08-04685-0.  Google Scholar

[11]

H. Helmholtz, Uber Integrale der hydrodynamischen Gleichungen, Welche den Wirbelbewegungen entsprechen, Crelle's Journal für Mathematik, 55 (1858), 25-55; English translation by P. G. Tait, On the integrals of the hydrodynamical equations which express vortex-motion, Philosophical Magazine, (1867), 485-551. Google Scholar

[12]

A. N. Jensen, Algorithmic Aspects of Gröbner Fans and Tropical Varieties, Ph.D. Thesis, Department of Mathematical Sciences, University of Aarhus, Denmark, 2007. Google Scholar

[13]

A. N. Jensen, Gfan, a software system for Gröbner fans and tropical varieties,, , ().   Google Scholar

[14]

J. Kulevich, G. E. Roberts and C. Smith, Finiteness in the planar restricted four-body problem, Qual. Theory Dyn. Sys., 8 (2009), 357-370. doi: 10.1007/s12346-010-0006-9.  Google Scholar

[15]

J. L. Lagrange, Essai Sur le Problème des Trois Corps, Œuvres, 1772. Google Scholar

[16]

P. W. Lindstrom, The number of planar central configurations is finite when $N-1$ mass positions are fixed, Trans. Amer. Math. Soc., 353 (2001), 291-311. doi: 10.1090/S0002-9947-00-02568-X.  Google Scholar

[17]

D. Maglagan and B. Sturmfels, Introduction to Tropical Geometry, Expected publication date May 2015. Google Scholar

[18]

R. Moeckel, On central configurations, Math. Z., 205 (1990), 499-517. doi: 10.1007/BF02571259.  Google Scholar

[19]

R. Moeckel, Relative equilibria with clusters of small masses, J. Dyn. Diff. Eq., 9 (1997), 507-533. doi: 10.1007/BF02219396.  Google Scholar

[20]

R. Moeckel, Generic finiteness for Dziobek configurations, Trans. Amer. Math. Soc., 353 (2001), 4673-4686. doi: 10.1090/S0002-9947-01-02828-8.  Google Scholar

[21]

F. R. Moulton, The straight line solutions of the problem of n bodies, Ann. of Math., 12 (1910), 1-17. doi: 10.2307/2007159.  Google Scholar

[22]

I. Newton, Philosophi Naturalis Principia Mathematica, Royal Society, London, 1687. Google Scholar

[23]

G. Roberts, A continuum of relative equilibria in the five-body problem, Phys. D, 127 (1999), 141-145. doi: 10.1016/S0167-2789(98)00315-7.  Google Scholar

[24]

S. Smale, Mathematical problems for the next century, Mathematical Intelligencer, 20 (1998), 7-15. doi: 10.1007/BF03025291.  Google Scholar

[25]

D. Speyer and B. Sturmfels, The tropical Grassmannian, Adv. Geom., 4 (2004), 389-411. doi: 10.1515/advg.2004.023.  Google Scholar

[26]

F. Tien, Recursion Formulas of Central Configurations, Thesis, University Of Minnesota, 1993.  Google Scholar

[27]

A. Wintner, The Analytical Foundations of Celestial Mechanics, Princeton Math. Series, 5, Princeton University Press, Princeton, NJ, 1941.  Google Scholar

[28]

Z. Xia, Central configurations with many small masses, J. Differential Equations, 91 (1991), 168-179. doi: 10.1016/0022-0396(91)90137-X.  Google Scholar

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