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...
Ausführliche Beschreibung
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| Autor*in: |
Fusco, G. [verfasserIn] |
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| Format: |
Artikel |
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| Sprache: |
Englisch |
| Erschienen: |
2010 |
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| Anmerkung: |
© Springer-Verlag 2010 |
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| Übergeordnetes Werk: |
Enthalten in: Inventiones mathematicae - Springer-Verlag, 1966, 185(2010), 2 vom: 16. Dez., Seite 283-332 |
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| Übergeordnetes Werk: |
volume:185 ; year:2010 ; number:2 ; day:16 ; month:12 ; pages:283-332 |
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| DOI / URN: |
10.1007/s00222-010-0306-3 |
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OLC205092349X |
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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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10.1007/s00222-010-0306-3 doi (DE-627)OLC205092349X (DE-He213)s00222-010-0306-3-p DE-627 ger DE-627 rakwb eng 510 VZ 510 VZ 17,1 ssgn SA 5940 VZ rvk SA 5940 VZ rvk Fusco, G. verfasserin aut Platonic polyhedra, topological constraints and periodic solutions of the classical N-body problem 2010 Text txt rdacontent ohne Hilfsmittel zu benutzen n rdamedia Band nc 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 Periodic Orbit Periodic Motion Rotation Group Periodic Sequence Gronchi, G. F. aut Negrini, P. aut Enthalten in Inventiones mathematicae Springer-Verlag, 1966 185(2010), 2 vom: 16. Dez., Seite 283-332 (DE-627)129077453 (DE-600)2921-X (DE-576)014409992 0020-9910 nnns volume:185 year:2010 number:2 day:16 month:12 pages:283-332 https://doi.org/10.1007/s00222-010-0306-3 lizenzpflichtig Volltext GBV_USEFLAG_A SYSFLAG_A GBV_OLC SSG-OLC-MAT SSG-OPC-MAT GBV_ILN_20 GBV_ILN_24 GBV_ILN_30 GBV_ILN_31 GBV_ILN_40 GBV_ILN_70 GBV_ILN_2002 GBV_ILN_2004 GBV_ILN_2007 GBV_ILN_2010 GBV_ILN_2012 GBV_ILN_2018 GBV_ILN_2030 GBV_ILN_2088 GBV_ILN_2409 GBV_ILN_4027 GBV_ILN_4126 GBV_ILN_4266 GBV_ILN_4277 GBV_ILN_4305 GBV_ILN_4317 GBV_ILN_4318 GBV_ILN_4323 GBV_ILN_4325 GBV_ILN_4700 SA 5940 SA 5940 AR 185 2010 2 16 12 283-332 |
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10.1007/s00222-010-0306-3 doi (DE-627)OLC205092349X (DE-He213)s00222-010-0306-3-p DE-627 ger DE-627 rakwb eng 510 VZ 510 VZ 17,1 ssgn SA 5940 VZ rvk SA 5940 VZ rvk Fusco, G. verfasserin aut Platonic polyhedra, topological constraints and periodic solutions of the classical N-body problem 2010 Text txt rdacontent ohne Hilfsmittel zu benutzen n rdamedia Band nc 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 Periodic Orbit Periodic Motion Rotation Group Periodic Sequence Gronchi, G. F. aut Negrini, P. aut Enthalten in Inventiones mathematicae Springer-Verlag, 1966 185(2010), 2 vom: 16. Dez., Seite 283-332 (DE-627)129077453 (DE-600)2921-X (DE-576)014409992 0020-9910 nnns volume:185 year:2010 number:2 day:16 month:12 pages:283-332 https://doi.org/10.1007/s00222-010-0306-3 lizenzpflichtig Volltext GBV_USEFLAG_A SYSFLAG_A GBV_OLC SSG-OLC-MAT SSG-OPC-MAT GBV_ILN_20 GBV_ILN_24 GBV_ILN_30 GBV_ILN_31 GBV_ILN_40 GBV_ILN_70 GBV_ILN_2002 GBV_ILN_2004 GBV_ILN_2007 GBV_ILN_2010 GBV_ILN_2012 GBV_ILN_2018 GBV_ILN_2030 GBV_ILN_2088 GBV_ILN_2409 GBV_ILN_4027 GBV_ILN_4126 GBV_ILN_4266 GBV_ILN_4277 GBV_ILN_4305 GBV_ILN_4317 GBV_ILN_4318 GBV_ILN_4323 GBV_ILN_4325 GBV_ILN_4700 SA 5940 SA 5940 AR 185 2010 2 16 12 283-332 |
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10.1007/s00222-010-0306-3 doi (DE-627)OLC205092349X (DE-He213)s00222-010-0306-3-p DE-627 ger DE-627 rakwb eng 510 VZ 510 VZ 17,1 ssgn SA 5940 VZ rvk SA 5940 VZ rvk Fusco, G. verfasserin aut Platonic polyhedra, topological constraints and periodic solutions of the classical N-body problem 2010 Text txt rdacontent ohne Hilfsmittel zu benutzen n rdamedia Band nc 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 Periodic Orbit Periodic Motion Rotation Group Periodic Sequence Gronchi, G. F. aut Negrini, P. aut Enthalten in Inventiones mathematicae Springer-Verlag, 1966 185(2010), 2 vom: 16. Dez., Seite 283-332 (DE-627)129077453 (DE-600)2921-X (DE-576)014409992 0020-9910 nnns volume:185 year:2010 number:2 day:16 month:12 pages:283-332 https://doi.org/10.1007/s00222-010-0306-3 lizenzpflichtig Volltext GBV_USEFLAG_A SYSFLAG_A GBV_OLC SSG-OLC-MAT SSG-OPC-MAT GBV_ILN_20 GBV_ILN_24 GBV_ILN_30 GBV_ILN_31 GBV_ILN_40 GBV_ILN_70 GBV_ILN_2002 GBV_ILN_2004 GBV_ILN_2007 GBV_ILN_2010 GBV_ILN_2012 GBV_ILN_2018 GBV_ILN_2030 GBV_ILN_2088 GBV_ILN_2409 GBV_ILN_4027 GBV_ILN_4126 GBV_ILN_4266 GBV_ILN_4277 GBV_ILN_4305 GBV_ILN_4317 GBV_ILN_4318 GBV_ILN_4323 GBV_ILN_4325 GBV_ILN_4700 SA 5940 SA 5940 AR 185 2010 2 16 12 283-332 |
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10.1007/s00222-010-0306-3 doi (DE-627)OLC205092349X (DE-He213)s00222-010-0306-3-p DE-627 ger DE-627 rakwb eng 510 VZ 510 VZ 17,1 ssgn SA 5940 VZ rvk SA 5940 VZ rvk Fusco, G. verfasserin aut Platonic polyhedra, topological constraints and periodic solutions of the classical N-body problem 2010 Text txt rdacontent ohne Hilfsmittel zu benutzen n rdamedia Band nc 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 Periodic Orbit Periodic Motion Rotation Group Periodic Sequence Gronchi, G. F. aut Negrini, P. aut Enthalten in Inventiones mathematicae Springer-Verlag, 1966 185(2010), 2 vom: 16. Dez., Seite 283-332 (DE-627)129077453 (DE-600)2921-X (DE-576)014409992 0020-9910 nnns volume:185 year:2010 number:2 day:16 month:12 pages:283-332 https://doi.org/10.1007/s00222-010-0306-3 lizenzpflichtig Volltext GBV_USEFLAG_A SYSFLAG_A GBV_OLC SSG-OLC-MAT SSG-OPC-MAT GBV_ILN_20 GBV_ILN_24 GBV_ILN_30 GBV_ILN_31 GBV_ILN_40 GBV_ILN_70 GBV_ILN_2002 GBV_ILN_2004 GBV_ILN_2007 GBV_ILN_2010 GBV_ILN_2012 GBV_ILN_2018 GBV_ILN_2030 GBV_ILN_2088 GBV_ILN_2409 GBV_ILN_4027 GBV_ILN_4126 GBV_ILN_4266 GBV_ILN_4277 GBV_ILN_4305 GBV_ILN_4317 GBV_ILN_4318 GBV_ILN_4323 GBV_ILN_4325 GBV_ILN_4700 SA 5940 SA 5940 AR 185 2010 2 16 12 283-332 |
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10.1007/s00222-010-0306-3 doi (DE-627)OLC205092349X (DE-He213)s00222-010-0306-3-p DE-627 ger DE-627 rakwb eng 510 VZ 510 VZ 17,1 ssgn SA 5940 VZ rvk SA 5940 VZ rvk Fusco, G. verfasserin aut Platonic polyhedra, topological constraints and periodic solutions of the classical N-body problem 2010 Text txt rdacontent ohne Hilfsmittel zu benutzen n rdamedia Band nc 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 Periodic Orbit Periodic Motion Rotation Group Periodic Sequence Gronchi, G. F. aut Negrini, P. aut Enthalten in Inventiones mathematicae Springer-Verlag, 1966 185(2010), 2 vom: 16. Dez., Seite 283-332 (DE-627)129077453 (DE-600)2921-X (DE-576)014409992 0020-9910 nnns volume:185 year:2010 number:2 day:16 month:12 pages:283-332 https://doi.org/10.1007/s00222-010-0306-3 lizenzpflichtig Volltext GBV_USEFLAG_A SYSFLAG_A GBV_OLC SSG-OLC-MAT SSG-OPC-MAT GBV_ILN_20 GBV_ILN_24 GBV_ILN_30 GBV_ILN_31 GBV_ILN_40 GBV_ILN_70 GBV_ILN_2002 GBV_ILN_2004 GBV_ILN_2007 GBV_ILN_2010 GBV_ILN_2012 GBV_ILN_2018 GBV_ILN_2030 GBV_ILN_2088 GBV_ILN_2409 GBV_ILN_4027 GBV_ILN_4126 GBV_ILN_4266 GBV_ILN_4277 GBV_ILN_4305 GBV_ILN_4317 GBV_ILN_4318 GBV_ILN_4323 GBV_ILN_4325 GBV_ILN_4700 SA 5940 SA 5940 AR 185 2010 2 16 12 283-332 |
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Platonic polyhedra, topological constraints and periodic solutions of the classical N-body problem |
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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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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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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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