Platonic polyhedra, topological constraints and periodic solutions of the classical N-body problem

Abstract We prove the existence of a number of smooth periodic motions u∗ of the classical Newtonian N-body problem which, up to a relabeling of the N particles, are invariant under the rotation group $\mathcal{R}$ of one of the five Platonic polyhedra. The number N coincides with the order $|\mathc...
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Autor*in:	Fusco, G. [verfasserIn] Gronchi, G. F. Negrini, P.

Format:	E-Artikel
Sprache:	Englisch

Erschienen:	2010

Schlagwörter:	Periodic Solution Periodic Orbit Periodic Motion Rotation Group Periodic Sequence

Anmerkung:	© Springer-Verlag 2010

Übergeordnetes Werk:	Enthalten in: Inventiones mathematicae - Berlin : Springer, 1966, 185(2010), 2 vom: 16. Dez., Seite 283-332
Übergeordnetes Werk:	volume:185 ; year:2010 ; number:2 ; day:16 ; month:12 ; pages:283-332

Links:	Volltext

DOI / URN:	10.1007/s00222-010-0306-3

Katalog-ID:	SPR002453177

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520			\|a Abstract We prove the existence of a number of smooth periodic motions u∗ of the classical Newtonian N-body problem which, up to a relabeling of the N particles, are invariant under the rotation group $\mathcal{R}$ of one of the five Platonic polyhedra. The number N coincides with the order $\|\mathcal{R}\|$ of $\mathcal{R}$ and the particles have all the same mass. Our approach is variational and u∗ is a minimizer of the Lagrangian action $\mathcal{A}$ on a suitable subset $\mathcal{K}$ of the H1T-periodic maps u:ℝ→$ ℝ^{3N} $. The set ${\mathcal {K}}$ is a cone and is determined by imposing on u both topological and symmetry constraints which are defined in terms of the rotation group $\mathcal{R}$. There exist infinitely many such cones ${\mathcal {K}}$, all with the property that ${\mathcal {A}}\|_{{\mathcal {K}}}$ is coercive. For a certain number of them, using level estimates and local deformations, we show that minimizers are free of collisions and therefore classical solutions of the N-body problem with a rich geometric–kinematic structure.
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700	1		\|a Negrini, P. \|4 aut
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912			\|a GBV_ILN_230
912			\|a GBV_ILN_250
912			\|a GBV_ILN_267
912			\|a GBV_ILN_281
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912			\|a GBV_ILN_370
912			\|a GBV_ILN_602
912			\|a GBV_ILN_636
912			\|a GBV_ILN_702
912			\|a GBV_ILN_2001
912			\|a GBV_ILN_2003
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912			\|a GBV_ILN_2005
912			\|a GBV_ILN_2006
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912			\|a GBV_ILN_2009
912			\|a GBV_ILN_2010
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allfields	10.1007/s00222-010-0306-3 doi (DE-627)SPR002453177 (SPR)s00222-010-0306-3-e DE-627 ger DE-627 rakwb eng Fusco, G. verfasserin aut Platonic polyhedra, topological constraints and periodic solutions of the classical N-body problem 2010 Text txt rdacontent Computermedien c rdamedia Online-Ressource cr rdacarrier © Springer-Verlag 2010 Abstract We prove the existence of a number of smooth periodic motions u∗ of the classical Newtonian N-body problem which, up to a relabeling of the N particles, are invariant under the rotation group $\mathcal{R}$ of one of the five Platonic polyhedra. The number N coincides with the order $\|\mathcal{R}\|$ of $\mathcal{R}$ and the particles have all the same mass. Our approach is variational and u∗ is a minimizer of the Lagrangian action $\mathcal{A}$ on a suitable subset $\mathcal{K}$ of the H1T-periodic maps u:ℝ→$ ℝ^{3N} $. The set ${\mathcal {K}}$ is a cone and is determined by imposing on u both topological and symmetry constraints which are defined in terms of the rotation group $\mathcal{R}$. There exist infinitely many such cones ${\mathcal {K}}$, all with the property that ${\mathcal {A}}\|_{{\mathcal {K}}}$ is coercive. For a certain number of them, using level estimates and local deformations, we show that minimizers are free of collisions and therefore classical solutions of the N-body problem with a rich geometric–kinematic structure. Periodic Solution (dpeaa)DE-He213 Periodic Orbit (dpeaa)DE-He213 Periodic Motion (dpeaa)DE-He213 Rotation Group (dpeaa)DE-He213 Periodic Sequence (dpeaa)DE-He213 Gronchi, G. F. aut Negrini, P. aut Enthalten in Inventiones mathematicae Berlin : Springer, 1966 185(2010), 2 vom: 16. Dez., Seite 283-332 (DE-627)235503525 (DE-600)1398341-6 1432-1297 nnns volume:185 year:2010 number:2 day:16 month:12 pages:283-332 https://dx.doi.org/10.1007/s00222-010-0306-3 lizenzpflichtig Volltext GBV_USEFLAG_A SYSFLAG_A GBV_SPRINGER GBV_ILN_11 GBV_ILN_20 GBV_ILN_22 GBV_ILN_23 GBV_ILN_24 GBV_ILN_31 GBV_ILN_32 GBV_ILN_39 GBV_ILN_40 GBV_ILN_60 GBV_ILN_62 GBV_ILN_63 GBV_ILN_69 GBV_ILN_70 GBV_ILN_73 GBV_ILN_74 GBV_ILN_90 GBV_ILN_95 GBV_ILN_100 GBV_ILN_105 GBV_ILN_110 GBV_ILN_120 GBV_ILN_138 GBV_ILN_150 GBV_ILN_151 GBV_ILN_152 GBV_ILN_161 GBV_ILN_170 GBV_ILN_171 GBV_ILN_187 GBV_ILN_206 GBV_ILN_213 GBV_ILN_224 GBV_ILN_230 GBV_ILN_250 GBV_ILN_267 GBV_ILN_281 GBV_ILN_285 GBV_ILN_293 GBV_ILN_370 GBV_ILN_602 GBV_ILN_636 GBV_ILN_702 GBV_ILN_2001 GBV_ILN_2003 GBV_ILN_2004 GBV_ILN_2005 GBV_ILN_2006 GBV_ILN_2007 GBV_ILN_2008 GBV_ILN_2009 GBV_ILN_2010 GBV_ILN_2011 GBV_ILN_2014 GBV_ILN_2015 GBV_ILN_2020 GBV_ILN_2021 GBV_ILN_2025 GBV_ILN_2026 GBV_ILN_2027 GBV_ILN_2031 GBV_ILN_2034 GBV_ILN_2037 GBV_ILN_2038 GBV_ILN_2039 GBV_ILN_2044 GBV_ILN_2048 GBV_ILN_2049 GBV_ILN_2050 GBV_ILN_2055 GBV_ILN_2056 GBV_ILN_2057 GBV_ILN_2059 GBV_ILN_2061 GBV_ILN_2064 GBV_ILN_2065 GBV_ILN_2068 GBV_ILN_2070 GBV_ILN_2086 GBV_ILN_2088 GBV_ILN_2093 GBV_ILN_2106 GBV_ILN_2107 GBV_ILN_2108 GBV_ILN_2110 GBV_ILN_2111 GBV_ILN_2112 GBV_ILN_2113 GBV_ILN_2116 GBV_ILN_2118 GBV_ILN_2119 GBV_ILN_2122 GBV_ILN_2129 GBV_ILN_2143 GBV_ILN_2144 GBV_ILN_2147 GBV_ILN_2148 GBV_ILN_2152 GBV_ILN_2153 GBV_ILN_2188 GBV_ILN_2190 GBV_ILN_2232 GBV_ILN_2336 GBV_ILN_2446 GBV_ILN_2470 GBV_ILN_2472 GBV_ILN_2507 GBV_ILN_2522 GBV_ILN_2548 GBV_ILN_4012 GBV_ILN_4035 GBV_ILN_4037 GBV_ILN_4046 GBV_ILN_4112 GBV_ILN_4125 GBV_ILN_4126 GBV_ILN_4242 GBV_ILN_4246 GBV_ILN_4249 GBV_ILN_4251 GBV_ILN_4305 GBV_ILN_4306 GBV_ILN_4307 GBV_ILN_4313 GBV_ILN_4322 GBV_ILN_4323 GBV_ILN_4324 GBV_ILN_4325 GBV_ILN_4326 GBV_ILN_4328 GBV_ILN_4333 GBV_ILN_4334 GBV_ILN_4335 GBV_ILN_4336 GBV_ILN_4338 GBV_ILN_4393 GBV_ILN_4700 AR 185 2010 2 16 12 283-332
spelling	10.1007/s00222-010-0306-3 doi (DE-627)SPR002453177 (SPR)s00222-010-0306-3-e DE-627 ger DE-627 rakwb eng Fusco, G. verfasserin aut Platonic polyhedra, topological constraints and periodic solutions of the classical N-body problem 2010 Text txt rdacontent Computermedien c rdamedia Online-Ressource cr rdacarrier © Springer-Verlag 2010 Abstract We prove the existence of a number of smooth periodic motions u∗ of the classical Newtonian N-body problem which, up to a relabeling of the N particles, are invariant under the rotation group $\mathcal{R}$ of one of the five Platonic polyhedra. The number N coincides with the order $\|\mathcal{R}\|$ of $\mathcal{R}$ and the particles have all the same mass. Our approach is variational and u∗ is a minimizer of the Lagrangian action $\mathcal{A}$ on a suitable subset $\mathcal{K}$ of the H1T-periodic maps u:ℝ→$ ℝ^{3N} $. The set ${\mathcal {K}}$ is a cone and is determined by imposing on u both topological and symmetry constraints which are defined in terms of the rotation group $\mathcal{R}$. There exist infinitely many such cones ${\mathcal {K}}$, all with the property that ${\mathcal {A}}\|_{{\mathcal {K}}}$ is coercive. For a certain number of them, using level estimates and local deformations, we show that minimizers are free of collisions and therefore classical solutions of the N-body problem with a rich geometric–kinematic structure. Periodic Solution (dpeaa)DE-He213 Periodic Orbit (dpeaa)DE-He213 Periodic Motion (dpeaa)DE-He213 Rotation Group (dpeaa)DE-He213 Periodic Sequence (dpeaa)DE-He213 Gronchi, G. F. aut Negrini, P. aut Enthalten in Inventiones mathematicae Berlin : Springer, 1966 185(2010), 2 vom: 16. Dez., Seite 283-332 (DE-627)235503525 (DE-600)1398341-6 1432-1297 nnns volume:185 year:2010 number:2 day:16 month:12 pages:283-332 https://dx.doi.org/10.1007/s00222-010-0306-3 lizenzpflichtig Volltext GBV_USEFLAG_A SYSFLAG_A GBV_SPRINGER GBV_ILN_11 GBV_ILN_20 GBV_ILN_22 GBV_ILN_23 GBV_ILN_24 GBV_ILN_31 GBV_ILN_32 GBV_ILN_39 GBV_ILN_40 GBV_ILN_60 GBV_ILN_62 GBV_ILN_63 GBV_ILN_69 GBV_ILN_70 GBV_ILN_73 GBV_ILN_74 GBV_ILN_90 GBV_ILN_95 GBV_ILN_100 GBV_ILN_105 GBV_ILN_110 GBV_ILN_120 GBV_ILN_138 GBV_ILN_150 GBV_ILN_151 GBV_ILN_152 GBV_ILN_161 GBV_ILN_170 GBV_ILN_171 GBV_ILN_187 GBV_ILN_206 GBV_ILN_213 GBV_ILN_224 GBV_ILN_230 GBV_ILN_250 GBV_ILN_267 GBV_ILN_281 GBV_ILN_285 GBV_ILN_293 GBV_ILN_370 GBV_ILN_602 GBV_ILN_636 GBV_ILN_702 GBV_ILN_2001 GBV_ILN_2003 GBV_ILN_2004 GBV_ILN_2005 GBV_ILN_2006 GBV_ILN_2007 GBV_ILN_2008 GBV_ILN_2009 GBV_ILN_2010 GBV_ILN_2011 GBV_ILN_2014 GBV_ILN_2015 GBV_ILN_2020 GBV_ILN_2021 GBV_ILN_2025 GBV_ILN_2026 GBV_ILN_2027 GBV_ILN_2031 GBV_ILN_2034 GBV_ILN_2037 GBV_ILN_2038 GBV_ILN_2039 GBV_ILN_2044 GBV_ILN_2048 GBV_ILN_2049 GBV_ILN_2050 GBV_ILN_2055 GBV_ILN_2056 GBV_ILN_2057 GBV_ILN_2059 GBV_ILN_2061 GBV_ILN_2064 GBV_ILN_2065 GBV_ILN_2068 GBV_ILN_2070 GBV_ILN_2086 GBV_ILN_2088 GBV_ILN_2093 GBV_ILN_2106 GBV_ILN_2107 GBV_ILN_2108 GBV_ILN_2110 GBV_ILN_2111 GBV_ILN_2112 GBV_ILN_2113 GBV_ILN_2116 GBV_ILN_2118 GBV_ILN_2119 GBV_ILN_2122 GBV_ILN_2129 GBV_ILN_2143 GBV_ILN_2144 GBV_ILN_2147 GBV_ILN_2148 GBV_ILN_2152 GBV_ILN_2153 GBV_ILN_2188 GBV_ILN_2190 GBV_ILN_2232 GBV_ILN_2336 GBV_ILN_2446 GBV_ILN_2470 GBV_ILN_2472 GBV_ILN_2507 GBV_ILN_2522 GBV_ILN_2548 GBV_ILN_4012 GBV_ILN_4035 GBV_ILN_4037 GBV_ILN_4046 GBV_ILN_4112 GBV_ILN_4125 GBV_ILN_4126 GBV_ILN_4242 GBV_ILN_4246 GBV_ILN_4249 GBV_ILN_4251 GBV_ILN_4305 GBV_ILN_4306 GBV_ILN_4307 GBV_ILN_4313 GBV_ILN_4322 GBV_ILN_4323 GBV_ILN_4324 GBV_ILN_4325 GBV_ILN_4326 GBV_ILN_4328 GBV_ILN_4333 GBV_ILN_4334 GBV_ILN_4335 GBV_ILN_4336 GBV_ILN_4338 GBV_ILN_4393 GBV_ILN_4700 AR 185 2010 2 16 12 283-332
allfields_unstemmed	10.1007/s00222-010-0306-3 doi (DE-627)SPR002453177 (SPR)s00222-010-0306-3-e DE-627 ger DE-627 rakwb eng Fusco, G. verfasserin aut Platonic polyhedra, topological constraints and periodic solutions of the classical N-body problem 2010 Text txt rdacontent Computermedien c rdamedia Online-Ressource cr rdacarrier © Springer-Verlag 2010 Abstract We prove the existence of a number of smooth periodic motions u∗ of the classical Newtonian N-body problem which, up to a relabeling of the N particles, are invariant under the rotation group $\mathcal{R}$ of one of the five Platonic polyhedra. The number N coincides with the order $\|\mathcal{R}\|$ of $\mathcal{R}$ and the particles have all the same mass. Our approach is variational and u∗ is a minimizer of the Lagrangian action $\mathcal{A}$ on a suitable subset $\mathcal{K}$ of the H1T-periodic maps u:ℝ→$ ℝ^{3N} $. The set ${\mathcal {K}}$ is a cone and is determined by imposing on u both topological and symmetry constraints which are defined in terms of the rotation group $\mathcal{R}$. There exist infinitely many such cones ${\mathcal {K}}$, all with the property that ${\mathcal {A}}\|_{{\mathcal {K}}}$ is coercive. For a certain number of them, using level estimates and local deformations, we show that minimizers are free of collisions and therefore classical solutions of the N-body problem with a rich geometric–kinematic structure. Periodic Solution (dpeaa)DE-He213 Periodic Orbit (dpeaa)DE-He213 Periodic Motion (dpeaa)DE-He213 Rotation Group (dpeaa)DE-He213 Periodic Sequence (dpeaa)DE-He213 Gronchi, G. F. aut Negrini, P. aut Enthalten in Inventiones mathematicae Berlin : Springer, 1966 185(2010), 2 vom: 16. Dez., Seite 283-332 (DE-627)235503525 (DE-600)1398341-6 1432-1297 nnns volume:185 year:2010 number:2 day:16 month:12 pages:283-332 https://dx.doi.org/10.1007/s00222-010-0306-3 lizenzpflichtig Volltext GBV_USEFLAG_A SYSFLAG_A GBV_SPRINGER GBV_ILN_11 GBV_ILN_20 GBV_ILN_22 GBV_ILN_23 GBV_ILN_24 GBV_ILN_31 GBV_ILN_32 GBV_ILN_39 GBV_ILN_40 GBV_ILN_60 GBV_ILN_62 GBV_ILN_63 GBV_ILN_69 GBV_ILN_70 GBV_ILN_73 GBV_ILN_74 GBV_ILN_90 GBV_ILN_95 GBV_ILN_100 GBV_ILN_105 GBV_ILN_110 GBV_ILN_120 GBV_ILN_138 GBV_ILN_150 GBV_ILN_151 GBV_ILN_152 GBV_ILN_161 GBV_ILN_170 GBV_ILN_171 GBV_ILN_187 GBV_ILN_206 GBV_ILN_213 GBV_ILN_224 GBV_ILN_230 GBV_ILN_250 GBV_ILN_267 GBV_ILN_281 GBV_ILN_285 GBV_ILN_293 GBV_ILN_370 GBV_ILN_602 GBV_ILN_636 GBV_ILN_702 GBV_ILN_2001 GBV_ILN_2003 GBV_ILN_2004 GBV_ILN_2005 GBV_ILN_2006 GBV_ILN_2007 GBV_ILN_2008 GBV_ILN_2009 GBV_ILN_2010 GBV_ILN_2011 GBV_ILN_2014 GBV_ILN_2015 GBV_ILN_2020 GBV_ILN_2021 GBV_ILN_2025 GBV_ILN_2026 GBV_ILN_2027 GBV_ILN_2031 GBV_ILN_2034 GBV_ILN_2037 GBV_ILN_2038 GBV_ILN_2039 GBV_ILN_2044 GBV_ILN_2048 GBV_ILN_2049 GBV_ILN_2050 GBV_ILN_2055 GBV_ILN_2056 GBV_ILN_2057 GBV_ILN_2059 GBV_ILN_2061 GBV_ILN_2064 GBV_ILN_2065 GBV_ILN_2068 GBV_ILN_2070 GBV_ILN_2086 GBV_ILN_2088 GBV_ILN_2093 GBV_ILN_2106 GBV_ILN_2107 GBV_ILN_2108 GBV_ILN_2110 GBV_ILN_2111 GBV_ILN_2112 GBV_ILN_2113 GBV_ILN_2116 GBV_ILN_2118 GBV_ILN_2119 GBV_ILN_2122 GBV_ILN_2129 GBV_ILN_2143 GBV_ILN_2144 GBV_ILN_2147 GBV_ILN_2148 GBV_ILN_2152 GBV_ILN_2153 GBV_ILN_2188 GBV_ILN_2190 GBV_ILN_2232 GBV_ILN_2336 GBV_ILN_2446 GBV_ILN_2470 GBV_ILN_2472 GBV_ILN_2507 GBV_ILN_2522 GBV_ILN_2548 GBV_ILN_4012 GBV_ILN_4035 GBV_ILN_4037 GBV_ILN_4046 GBV_ILN_4112 GBV_ILN_4125 GBV_ILN_4126 GBV_ILN_4242 GBV_ILN_4246 GBV_ILN_4249 GBV_ILN_4251 GBV_ILN_4305 GBV_ILN_4306 GBV_ILN_4307 GBV_ILN_4313 GBV_ILN_4322 GBV_ILN_4323 GBV_ILN_4324 GBV_ILN_4325 GBV_ILN_4326 GBV_ILN_4328 GBV_ILN_4333 GBV_ILN_4334 GBV_ILN_4335 GBV_ILN_4336 GBV_ILN_4338 GBV_ILN_4393 GBV_ILN_4700 AR 185 2010 2 16 12 283-332
allfieldsGer	10.1007/s00222-010-0306-3 doi (DE-627)SPR002453177 (SPR)s00222-010-0306-3-e DE-627 ger DE-627 rakwb eng Fusco, G. verfasserin aut Platonic polyhedra, topological constraints and periodic solutions of the classical N-body problem 2010 Text txt rdacontent Computermedien c rdamedia Online-Ressource cr rdacarrier © Springer-Verlag 2010 Abstract We prove the existence of a number of smooth periodic motions u∗ of the classical Newtonian N-body problem which, up to a relabeling of the N particles, are invariant under the rotation group $\mathcal{R}$ of one of the five Platonic polyhedra. The number N coincides with the order $\|\mathcal{R}\|$ of $\mathcal{R}$ and the particles have all the same mass. Our approach is variational and u∗ is a minimizer of the Lagrangian action $\mathcal{A}$ on a suitable subset $\mathcal{K}$ of the H1T-periodic maps u:ℝ→$ ℝ^{3N} $. The set ${\mathcal {K}}$ is a cone and is determined by imposing on u both topological and symmetry constraints which are defined in terms of the rotation group $\mathcal{R}$. There exist infinitely many such cones ${\mathcal {K}}$, all with the property that ${\mathcal {A}}\|_{{\mathcal {K}}}$ is coercive. For a certain number of them, using level estimates and local deformations, we show that minimizers are free of collisions and therefore classical solutions of the N-body problem with a rich geometric–kinematic structure. Periodic Solution (dpeaa)DE-He213 Periodic Orbit (dpeaa)DE-He213 Periodic Motion (dpeaa)DE-He213 Rotation Group (dpeaa)DE-He213 Periodic Sequence (dpeaa)DE-He213 Gronchi, G. F. aut Negrini, P. aut Enthalten in Inventiones mathematicae Berlin : Springer, 1966 185(2010), 2 vom: 16. Dez., Seite 283-332 (DE-627)235503525 (DE-600)1398341-6 1432-1297 nnns volume:185 year:2010 number:2 day:16 month:12 pages:283-332 https://dx.doi.org/10.1007/s00222-010-0306-3 lizenzpflichtig Volltext GBV_USEFLAG_A SYSFLAG_A GBV_SPRINGER GBV_ILN_11 GBV_ILN_20 GBV_ILN_22 GBV_ILN_23 GBV_ILN_24 GBV_ILN_31 GBV_ILN_32 GBV_ILN_39 GBV_ILN_40 GBV_ILN_60 GBV_ILN_62 GBV_ILN_63 GBV_ILN_69 GBV_ILN_70 GBV_ILN_73 GBV_ILN_74 GBV_ILN_90 GBV_ILN_95 GBV_ILN_100 GBV_ILN_105 GBV_ILN_110 GBV_ILN_120 GBV_ILN_138 GBV_ILN_150 GBV_ILN_151 GBV_ILN_152 GBV_ILN_161 GBV_ILN_170 GBV_ILN_171 GBV_ILN_187 GBV_ILN_206 GBV_ILN_213 GBV_ILN_224 GBV_ILN_230 GBV_ILN_250 GBV_ILN_267 GBV_ILN_281 GBV_ILN_285 GBV_ILN_293 GBV_ILN_370 GBV_ILN_602 GBV_ILN_636 GBV_ILN_702 GBV_ILN_2001 GBV_ILN_2003 GBV_ILN_2004 GBV_ILN_2005 GBV_ILN_2006 GBV_ILN_2007 GBV_ILN_2008 GBV_ILN_2009 GBV_ILN_2010 GBV_ILN_2011 GBV_ILN_2014 GBV_ILN_2015 GBV_ILN_2020 GBV_ILN_2021 GBV_ILN_2025 GBV_ILN_2026 GBV_ILN_2027 GBV_ILN_2031 GBV_ILN_2034 GBV_ILN_2037 GBV_ILN_2038 GBV_ILN_2039 GBV_ILN_2044 GBV_ILN_2048 GBV_ILN_2049 GBV_ILN_2050 GBV_ILN_2055 GBV_ILN_2056 GBV_ILN_2057 GBV_ILN_2059 GBV_ILN_2061 GBV_ILN_2064 GBV_ILN_2065 GBV_ILN_2068 GBV_ILN_2070 GBV_ILN_2086 GBV_ILN_2088 GBV_ILN_2093 GBV_ILN_2106 GBV_ILN_2107 GBV_ILN_2108 GBV_ILN_2110 GBV_ILN_2111 GBV_ILN_2112 GBV_ILN_2113 GBV_ILN_2116 GBV_ILN_2118 GBV_ILN_2119 GBV_ILN_2122 GBV_ILN_2129 GBV_ILN_2143 GBV_ILN_2144 GBV_ILN_2147 GBV_ILN_2148 GBV_ILN_2152 GBV_ILN_2153 GBV_ILN_2188 GBV_ILN_2190 GBV_ILN_2232 GBV_ILN_2336 GBV_ILN_2446 GBV_ILN_2470 GBV_ILN_2472 GBV_ILN_2507 GBV_ILN_2522 GBV_ILN_2548 GBV_ILN_4012 GBV_ILN_4035 GBV_ILN_4037 GBV_ILN_4046 GBV_ILN_4112 GBV_ILN_4125 GBV_ILN_4126 GBV_ILN_4242 GBV_ILN_4246 GBV_ILN_4249 GBV_ILN_4251 GBV_ILN_4305 GBV_ILN_4306 GBV_ILN_4307 GBV_ILN_4313 GBV_ILN_4322 GBV_ILN_4323 GBV_ILN_4324 GBV_ILN_4325 GBV_ILN_4326 GBV_ILN_4328 GBV_ILN_4333 GBV_ILN_4334 GBV_ILN_4335 GBV_ILN_4336 GBV_ILN_4338 GBV_ILN_4393 GBV_ILN_4700 AR 185 2010 2 16 12 283-332
allfieldsSound	10.1007/s00222-010-0306-3 doi (DE-627)SPR002453177 (SPR)s00222-010-0306-3-e DE-627 ger DE-627 rakwb eng Fusco, G. verfasserin aut Platonic polyhedra, topological constraints and periodic solutions of the classical N-body problem 2010 Text txt rdacontent Computermedien c rdamedia Online-Ressource cr rdacarrier © Springer-Verlag 2010 Abstract We prove the existence of a number of smooth periodic motions u∗ of the classical Newtonian N-body problem which, up to a relabeling of the N particles, are invariant under the rotation group $\mathcal{R}$ of one of the five Platonic polyhedra. The number N coincides with the order $\|\mathcal{R}\|$ of $\mathcal{R}$ and the particles have all the same mass. Our approach is variational and u∗ is a minimizer of the Lagrangian action $\mathcal{A}$ on a suitable subset $\mathcal{K}$ of the H1T-periodic maps u:ℝ→$ ℝ^{3N} $. The set ${\mathcal {K}}$ is a cone and is determined by imposing on u both topological and symmetry constraints which are defined in terms of the rotation group $\mathcal{R}$. There exist infinitely many such cones ${\mathcal {K}}$, all with the property that ${\mathcal {A}}\|_{{\mathcal {K}}}$ is coercive. For a certain number of them, using level estimates and local deformations, we show that minimizers are free of collisions and therefore classical solutions of the N-body problem with a rich geometric–kinematic structure. Periodic Solution (dpeaa)DE-He213 Periodic Orbit (dpeaa)DE-He213 Periodic Motion (dpeaa)DE-He213 Rotation Group (dpeaa)DE-He213 Periodic Sequence (dpeaa)DE-He213 Gronchi, G. F. aut Negrini, P. aut Enthalten in Inventiones mathematicae Berlin : Springer, 1966 185(2010), 2 vom: 16. Dez., Seite 283-332 (DE-627)235503525 (DE-600)1398341-6 1432-1297 nnns volume:185 year:2010 number:2 day:16 month:12 pages:283-332 https://dx.doi.org/10.1007/s00222-010-0306-3 lizenzpflichtig Volltext GBV_USEFLAG_A SYSFLAG_A GBV_SPRINGER GBV_ILN_11 GBV_ILN_20 GBV_ILN_22 GBV_ILN_23 GBV_ILN_24 GBV_ILN_31 GBV_ILN_32 GBV_ILN_39 GBV_ILN_40 GBV_ILN_60 GBV_ILN_62 GBV_ILN_63 GBV_ILN_69 GBV_ILN_70 GBV_ILN_73 GBV_ILN_74 GBV_ILN_90 GBV_ILN_95 GBV_ILN_100 GBV_ILN_105 GBV_ILN_110 GBV_ILN_120 GBV_ILN_138 GBV_ILN_150 GBV_ILN_151 GBV_ILN_152 GBV_ILN_161 GBV_ILN_170 GBV_ILN_171 GBV_ILN_187 GBV_ILN_206 GBV_ILN_213 GBV_ILN_224 GBV_ILN_230 GBV_ILN_250 GBV_ILN_267 GBV_ILN_281 GBV_ILN_285 GBV_ILN_293 GBV_ILN_370 GBV_ILN_602 GBV_ILN_636 GBV_ILN_702 GBV_ILN_2001 GBV_ILN_2003 GBV_ILN_2004 GBV_ILN_2005 GBV_ILN_2006 GBV_ILN_2007 GBV_ILN_2008 GBV_ILN_2009 GBV_ILN_2010 GBV_ILN_2011 GBV_ILN_2014 GBV_ILN_2015 GBV_ILN_2020 GBV_ILN_2021 GBV_ILN_2025 GBV_ILN_2026 GBV_ILN_2027 GBV_ILN_2031 GBV_ILN_2034 GBV_ILN_2037 GBV_ILN_2038 GBV_ILN_2039 GBV_ILN_2044 GBV_ILN_2048 GBV_ILN_2049 GBV_ILN_2050 GBV_ILN_2055 GBV_ILN_2056 GBV_ILN_2057 GBV_ILN_2059 GBV_ILN_2061 GBV_ILN_2064 GBV_ILN_2065 GBV_ILN_2068 GBV_ILN_2070 GBV_ILN_2086 GBV_ILN_2088 GBV_ILN_2093 GBV_ILN_2106 GBV_ILN_2107 GBV_ILN_2108 GBV_ILN_2110 GBV_ILN_2111 GBV_ILN_2112 GBV_ILN_2113 GBV_ILN_2116 GBV_ILN_2118 GBV_ILN_2119 GBV_ILN_2122 GBV_ILN_2129 GBV_ILN_2143 GBV_ILN_2144 GBV_ILN_2147 GBV_ILN_2148 GBV_ILN_2152 GBV_ILN_2153 GBV_ILN_2188 GBV_ILN_2190 GBV_ILN_2232 GBV_ILN_2336 GBV_ILN_2446 GBV_ILN_2470 GBV_ILN_2472 GBV_ILN_2507 GBV_ILN_2522 GBV_ILN_2548 GBV_ILN_4012 GBV_ILN_4035 GBV_ILN_4037 GBV_ILN_4046 GBV_ILN_4112 GBV_ILN_4125 GBV_ILN_4126 GBV_ILN_4242 GBV_ILN_4246 GBV_ILN_4249 GBV_ILN_4251 GBV_ILN_4305 GBV_ILN_4306 GBV_ILN_4307 GBV_ILN_4313 GBV_ILN_4322 GBV_ILN_4323 GBV_ILN_4324 GBV_ILN_4325 GBV_ILN_4326 GBV_ILN_4328 GBV_ILN_4333 GBV_ILN_4334 GBV_ILN_4335 GBV_ILN_4336 GBV_ILN_4338 GBV_ILN_4393 GBV_ILN_4700 AR 185 2010 2 16 12 283-332
language	English
source	Enthalten in Inventiones mathematicae 185(2010), 2 vom: 16. Dez., Seite 283-332 volume:185 year:2010 number:2 day:16 month:12 pages:283-332
sourceStr	Enthalten in Inventiones mathematicae 185(2010), 2 vom: 16. Dez., Seite 283-332 volume:185 year:2010 number:2 day:16 month:12 pages:283-332
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topic_facet	Periodic Solution Periodic Orbit Periodic Motion Rotation Group Periodic Sequence
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container_title	Inventiones mathematicae
authorswithroles_txt_mv	Fusco, G. @@aut@@ Gronchi, G. F. @@aut@@ Negrini, P. @@aut@@
publishDateDaySort_date	2010-12-16T00:00:00Z
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The number N coincides with the order $\|\mathcal{R}\|$ of $\mathcal{R}$ and the particles have all the same mass. Our approach is variational and u∗ is a minimizer of the Lagrangian action $\mathcal{A}$ on a suitable subset $\mathcal{K}$ of the H1T-periodic maps u:ℝ→$ ℝ^{3N} $. The set ${\mathcal {K}}$ is a cone and is determined by imposing on u both topological and symmetry constraints which are defined in terms of the rotation group $\mathcal{R}$. There exist infinitely many such cones ${\mathcal {K}}$, all with the property that ${\mathcal {A}}\|_{{\mathcal {K}}}$ is coercive. 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author	Fusco, G.
spellingShingle	Fusco, G. misc Periodic Solution misc Periodic Orbit misc Periodic Motion misc Rotation Group misc Periodic Sequence Platonic polyhedra, topological constraints and periodic solutions of the classical N-body problem
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topic_title	Platonic polyhedra, topological constraints and periodic solutions of the classical N-body problem Periodic Solution (dpeaa)DE-He213 Periodic Orbit (dpeaa)DE-He213 Periodic Motion (dpeaa)DE-He213 Rotation Group (dpeaa)DE-He213 Periodic Sequence (dpeaa)DE-He213
topic	misc Periodic Solution misc Periodic Orbit misc Periodic Motion misc Rotation Group misc Periodic Sequence
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title_full	Platonic polyhedra, topological constraints and periodic solutions of the classical N-body problem
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title_sort	platonic polyhedra, topological constraints and periodic solutions of the classical n-body problem
title_auth	Platonic polyhedra, topological constraints and periodic solutions of the classical N-body problem
abstract	Abstract We prove the existence of a number of smooth periodic motions u∗ of the classical Newtonian N-body problem which, up to a relabeling of the N particles, are invariant under the rotation group $\mathcal{R}$ of one of the five Platonic polyhedra. The number N coincides with the order $\|\mathcal{R}\|$ of $\mathcal{R}$ and the particles have all the same mass. Our approach is variational and u∗ is a minimizer of the Lagrangian action $\mathcal{A}$ on a suitable subset $\mathcal{K}$ of the H1T-periodic maps u:ℝ→$ ℝ^{3N} $. The set ${\mathcal {K}}$ is a cone and is determined by imposing on u both topological and symmetry constraints which are defined in terms of the rotation group $\mathcal{R}$. There exist infinitely many such cones ${\mathcal {K}}$, all with the property that ${\mathcal {A}}\|_{{\mathcal {K}}}$ is coercive. For a certain number of them, using level estimates and local deformations, we show that minimizers are free of collisions and therefore classical solutions of the N-body problem with a rich geometric–kinematic structure. © Springer-Verlag 2010
abstractGer	Abstract We prove the existence of a number of smooth periodic motions u∗ of the classical Newtonian N-body problem which, up to a relabeling of the N particles, are invariant under the rotation group $\mathcal{R}$ of one of the five Platonic polyhedra. The number N coincides with the order $\|\mathcal{R}\|$ of $\mathcal{R}$ and the particles have all the same mass. Our approach is variational and u∗ is a minimizer of the Lagrangian action $\mathcal{A}$ on a suitable subset $\mathcal{K}$ of the H1T-periodic maps u:ℝ→$ ℝ^{3N} $. The set ${\mathcal {K}}$ is a cone and is determined by imposing on u both topological and symmetry constraints which are defined in terms of the rotation group $\mathcal{R}$. There exist infinitely many such cones ${\mathcal {K}}$, all with the property that ${\mathcal {A}}\|_{{\mathcal {K}}}$ is coercive. For a certain number of them, using level estimates and local deformations, we show that minimizers are free of collisions and therefore classical solutions of the N-body problem with a rich geometric–kinematic structure. © Springer-Verlag 2010
abstract_unstemmed	Abstract We prove the existence of a number of smooth periodic motions u∗ of the classical Newtonian N-body problem which, up to a relabeling of the N particles, are invariant under the rotation group $\mathcal{R}$ of one of the five Platonic polyhedra. The number N coincides with the order $\|\mathcal{R}\|$ of $\mathcal{R}$ and the particles have all the same mass. Our approach is variational and u∗ is a minimizer of the Lagrangian action $\mathcal{A}$ on a suitable subset $\mathcal{K}$ of the H1T-periodic maps u:ℝ→$ ℝ^{3N} $. The set ${\mathcal {K}}$ is a cone and is determined by imposing on u both topological and symmetry constraints which are defined in terms of the rotation group $\mathcal{R}$. There exist infinitely many such cones ${\mathcal {K}}$, all with the property that ${\mathcal {A}}\|_{{\mathcal {K}}}$ is coercive. For a certain number of them, using level estimates and local deformations, we show that minimizers are free of collisions and therefore classical solutions of the N-body problem with a rich geometric–kinematic structure. © Springer-Verlag 2010
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title_short	Platonic polyhedra, topological constraints and periodic solutions of the classical N-body problem
url	https://dx.doi.org/10.1007/s00222-010-0306-3
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author2	Gronchi, G. F. Negrini, P.
author2Str	Gronchi, G. F. Negrini, P.
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doi_str	10.1007/s00222-010-0306-3
up_date	2024-07-04T03:03:52.508Z
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Platonic polyhedra, topological constraints and periodic solutions of the classical N-body problem

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