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Finiteness of relative equilibria in the planar generalized $N$-body problem with fixed subconfigurations
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 |
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. |
[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. |
[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. |
[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. |
[5] |
J. Chazy, Sur certaines trajectoires du problème des n corps, Bull. Astron., 35 (1918), 321-389. |
[6] |
L. Euler, De motu rectilineo trium corporum se mutuo attrahentium, Novi Comm. Acad. Sci. Imp. Petrop., 11 (1767), 144-151. |
[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. |
[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. |
[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. |
[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. |
[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. |
[12] |
A. N. Jensen, Algorithmic Aspects of Gröbner Fans and Tropical Varieties, Ph.D. Thesis, Department of Mathematical Sciences, University of Aarhus, Denmark, 2007. |
[13] |
A. N. Jensen, Gfan, a software system for Gröbner fans and tropical varieties,, , ().
|
[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. |
[15] |
J. L. Lagrange, Essai Sur le Problème des Trois Corps, Œuvres, 1772. |
[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. |
[17] |
D. Maglagan and B. Sturmfels, Introduction to Tropical Geometry, Expected publication date May 2015. |
[18] |
R. Moeckel, On central configurations, Math. Z., 205 (1990), 499-517.
doi: 10.1007/BF02571259. |
[19] |
R. Moeckel, Relative equilibria with clusters of small masses, J. Dyn. Diff. Eq., 9 (1997), 507-533.
doi: 10.1007/BF02219396. |
[20] |
R. Moeckel, Generic finiteness for Dziobek configurations, Trans. Amer. Math. Soc., 353 (2001), 4673-4686.
doi: 10.1090/S0002-9947-01-02828-8. |
[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. |
[22] |
I. Newton, Philosophi Naturalis Principia Mathematica, Royal Society, London, 1687. |
[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. |
[24] |
S. Smale, Mathematical problems for the next century, Mathematical Intelligencer, 20 (1998), 7-15.
doi: 10.1007/BF03025291. |
[25] |
D. Speyer and B. Sturmfels, The tropical Grassmannian, Adv. Geom., 4 (2004), 389-411.
doi: 10.1515/advg.2004.023. |
[26] |
F. Tien, Recursion Formulas of Central Configurations, Thesis, University Of Minnesota, 1993. |
[27] |
A. Wintner, The Analytical Foundations of Celestial Mechanics, Princeton Math. Series, 5, Princeton University Press, Princeton, NJ, 1941. |
[28] |
Z. Xia, Central configurations with many small masses, J. Differential Equations, 91 (1991), 168-179.
doi: 10.1016/0022-0396(91)90137-X. |
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. |
[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. |
[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. |
[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. |
[5] |
J. Chazy, Sur certaines trajectoires du problème des n corps, Bull. Astron., 35 (1918), 321-389. |
[6] |
L. Euler, De motu rectilineo trium corporum se mutuo attrahentium, Novi Comm. Acad. Sci. Imp. Petrop., 11 (1767), 144-151. |
[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. |
[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. |
[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. |
[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. |
[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. |
[12] |
A. N. Jensen, Algorithmic Aspects of Gröbner Fans and Tropical Varieties, Ph.D. Thesis, Department of Mathematical Sciences, University of Aarhus, Denmark, 2007. |
[13] |
A. N. Jensen, Gfan, a software system for Gröbner fans and tropical varieties,, , ().
|
[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. |
[15] |
J. L. Lagrange, Essai Sur le Problème des Trois Corps, Œuvres, 1772. |
[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. |
[17] |
D. Maglagan and B. Sturmfels, Introduction to Tropical Geometry, Expected publication date May 2015. |
[18] |
R. Moeckel, On central configurations, Math. Z., 205 (1990), 499-517.
doi: 10.1007/BF02571259. |
[19] |
R. Moeckel, Relative equilibria with clusters of small masses, J. Dyn. Diff. Eq., 9 (1997), 507-533.
doi: 10.1007/BF02219396. |
[20] |
R. Moeckel, Generic finiteness for Dziobek configurations, Trans. Amer. Math. Soc., 353 (2001), 4673-4686.
doi: 10.1090/S0002-9947-01-02828-8. |
[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. |
[22] |
I. Newton, Philosophi Naturalis Principia Mathematica, Royal Society, London, 1687. |
[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. |
[24] |
S. Smale, Mathematical problems for the next century, Mathematical Intelligencer, 20 (1998), 7-15.
doi: 10.1007/BF03025291. |
[25] |
D. Speyer and B. Sturmfels, The tropical Grassmannian, Adv. Geom., 4 (2004), 389-411.
doi: 10.1515/advg.2004.023. |
[26] |
F. Tien, Recursion Formulas of Central Configurations, Thesis, University Of Minnesota, 1993. |
[27] |
A. Wintner, The Analytical Foundations of Celestial Mechanics, Princeton Math. Series, 5, Princeton University Press, Princeton, NJ, 1941. |
[28] |
Z. Xia, Central configurations with many small masses, J. Differential Equations, 91 (1991), 168-179.
doi: 10.1016/0022-0396(91)90137-X. |
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