Convexity Methods in Hamiltonian Mechanics:
Gespeichert in:
| 1. Verfasser: | |
|---|---|
| Format: | Elektronisch E-Book |
| Sprache: | English |
| Veröffentlicht: |
Berlin, Heidelberg
Springer Berlin Heidelberg
1990
|
| Schriftenreihe: | Ergebnisse der Mathematik und ihrer Grenzgebiete, A Series of Modern Surveys in Mathematics
19 |
| Schlagworte: | |
| Online-Zugang: | Volltext |
| Beschreibung: | In the case of completely integrable systems, periodic solutions are found by inspection. For nonintegrable systems, such as the three-body problem in celestial mechanics, they are found by perturbation theory: there is a small parameter € in the problem, the mass of the perturbing body for instance, and for € = 0 the system becomes completely integrable. One then tries to show that its periodic solutions will subsist for € -# 0 small enough. Poincare also introduced global methods, relying on the topological properties of the flow, and the fact that it preserves the 2-form L~=l dPi 1\ dqi' The most celebrated result he obtained in this direction is his last geometric theorem, which states that an area-preserving map of the annulus which rotates the inner circle and the outer circle in opposite directions must have two fixed points. And now another ancient theme appear: the least action principle. It states that the periodic solutions of a Hamiltonian system are extremals of a suitable integral over closed curves. In other words, the problem is variational. This fact was known to Fermat, and Maupertuis put it in the Hamiltonian formalism. In spite of its great aesthetic appeal, the least action principle has had little impact in Hamiltonian mechanics. There is, of course, one exception, Emmy Noether's theorem, which relates integrals ofthe motion to symmetries of the equations. But until recently, no periodic solution had ever been found by variational methods |
| Beschreibung: | 1 Online-Ressource (X, 247p. 4 illus) |
| ISBN: | 9783642743313 9783642743337 |
| ISSN: | 0071-1136 |
| DOI: | 10.1007/978-3-642-74331-3 |
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| 500 | |a In the case of completely integrable systems, periodic solutions are found by inspection. For nonintegrable systems, such as the three-body problem in celestial mechanics, they are found by perturbation theory: there is a small parameter € in the problem, the mass of the perturbing body for instance, and for € = 0 the system becomes completely integrable. One then tries to show that its periodic solutions will subsist for € -# 0 small enough. Poincare also introduced global methods, relying on the topological properties of the flow, and the fact that it preserves the 2-form L~=l dPi 1\ dqi' The most celebrated result he obtained in this direction is his last geometric theorem, which states that an area-preserving map of the annulus which rotates the inner circle and the outer circle in opposite directions must have two fixed points. And now another ancient theme appear: the least action principle. It states that the periodic solutions of a Hamiltonian system are extremals of a suitable integral over closed curves. In other words, the problem is variational. This fact was known to Fermat, and Maupertuis put it in the Hamiltonian formalism. In spite of its great aesthetic appeal, the least action principle has had little impact in Hamiltonian mechanics. There is, of course, one exception, Emmy Noether's theorem, which relates integrals ofthe motion to symmetries of the equations. But until recently, no periodic solution had ever been found by variational methods | ||
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Datensatz im Suchindex
| _version_ | 1804153098068819968 |
|---|---|
| any_adam_object | |
| author | Ekeland, Ivar |
| author_facet | Ekeland, Ivar |
| author_role | aut |
| author_sort | Ekeland, Ivar |
| author_variant | i e ie |
| building | Verbundindex |
| bvnumber | BV042422970 |
| classification_tum | MAT 000 |
| collection | ZDB-2-SMA ZDB-2-BAE |
| ctrlnum | (OCoLC)1184495632 (DE-599)BVBBV042422970 |
| dewey-full | 515.353 |
| dewey-hundreds | 500 - Natural sciences and mathematics |
| dewey-ones | 515 - Analysis |
| dewey-raw | 515.353 |
| dewey-search | 515.353 |
| dewey-sort | 3515.353 |
| dewey-tens | 510 - Mathematics |
| discipline | Mathematik |
| doi_str_mv | 10.1007/978-3-642-74331-3 |
| format | Electronic eBook |
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| id | DE-604.BV042422970 |
| illustrated | Not Illustrated |
| indexdate | 2024-07-10T01:21:12Z |
| institution | BVB |
| isbn | 9783642743313 9783642743337 |
| issn | 0071-1136 |
| language | English |
| oai_aleph_id | oai:aleph.bib-bvb.de:BVB01-027858387 |
| oclc_num | 1184495632 |
| open_access_boolean | |
| owner | DE-384 DE-703 DE-91 DE-BY-TUM DE-634 |
| owner_facet | DE-384 DE-703 DE-91 DE-BY-TUM DE-634 |
| physical | 1 Online-Ressource (X, 247p. 4 illus) |
| psigel | ZDB-2-SMA ZDB-2-BAE ZDB-2-SMA_Archive |
| publishDate | 1990 |
| publishDateSearch | 1990 |
| publishDateSort | 1990 |
| publisher | Springer Berlin Heidelberg |
| record_format | marc |
| series2 | Ergebnisse der Mathematik und ihrer Grenzgebiete, A Series of Modern Surveys in Mathematics |
| spelling | Ekeland, Ivar Verfasser aut Convexity Methods in Hamiltonian Mechanics by Ivar Ekeland Berlin, Heidelberg Springer Berlin Heidelberg 1990 1 Online-Ressource (X, 247p. 4 illus) txt rdacontent c rdamedia cr rdacarrier Ergebnisse der Mathematik und ihrer Grenzgebiete, A Series of Modern Surveys in Mathematics 19 0071-1136 In the case of completely integrable systems, periodic solutions are found by inspection. For nonintegrable systems, such as the three-body problem in celestial mechanics, they are found by perturbation theory: there is a small parameter € in the problem, the mass of the perturbing body for instance, and for € = 0 the system becomes completely integrable. One then tries to show that its periodic solutions will subsist for € -# 0 small enough. Poincare also introduced global methods, relying on the topological properties of the flow, and the fact that it preserves the 2-form L~=l dPi 1\ dqi' The most celebrated result he obtained in this direction is his last geometric theorem, which states that an area-preserving map of the annulus which rotates the inner circle and the outer circle in opposite directions must have two fixed points. And now another ancient theme appear: the least action principle. It states that the periodic solutions of a Hamiltonian system are extremals of a suitable integral over closed curves. In other words, the problem is variational. This fact was known to Fermat, and Maupertuis put it in the Hamiltonian formalism. In spite of its great aesthetic appeal, the least action principle has had little impact in Hamiltonian mechanics. There is, of course, one exception, Emmy Noether's theorem, which relates integrals ofthe motion to symmetries of the equations. But until recently, no periodic solution had ever been found by variational methods Mathematics Differential equations, partial Mathematical optimization Partial Differential Equations Calculus of Variations and Optimal Control; Optimization Theoretical, Mathematical and Computational Physics Game Theory, Economics, Social and Behav. Sciences Mathematik Mechanik (DE-588)4038168-7 gnd rswk-swf Hamiltonsches System (DE-588)4139943-2 gnd rswk-swf Konvexe Analysis (DE-588)4138566-4 gnd rswk-swf Konvexe Menge (DE-588)4165212-5 gnd rswk-swf Hamilton-Funktion (DE-588)4323257-7 gnd rswk-swf Konvexe Funktion (DE-588)4139679-0 gnd rswk-swf Theoretische Mechanik (DE-588)4185100-6 gnd rswk-swf Konvexität (DE-588)4114284-6 gnd rswk-swf Variationsproblem (DE-588)4187419-5 gnd rswk-swf Hamiltonsches System (DE-588)4139943-2 s Variationsproblem (DE-588)4187419-5 s 1\p DE-604 Konvexität (DE-588)4114284-6 s 2\p DE-604 Hamilton-Funktion (DE-588)4323257-7 s Konvexe Funktion (DE-588)4139679-0 s 3\p DE-604 Theoretische Mechanik (DE-588)4185100-6 s 4\p DE-604 Mechanik (DE-588)4038168-7 s Konvexe Menge (DE-588)4165212-5 s 5\p DE-604 Konvexe Analysis (DE-588)4138566-4 s 6\p DE-604 https://doi.org/10.1007/978-3-642-74331-3 Verlag Volltext 1\p cgwrk 20201028 DE-101 https://d-nb.info/provenance/plan#cgwrk 2\p cgwrk 20201028 DE-101 https://d-nb.info/provenance/plan#cgwrk 3\p cgwrk 20201028 DE-101 https://d-nb.info/provenance/plan#cgwrk 4\p cgwrk 20201028 DE-101 https://d-nb.info/provenance/plan#cgwrk 5\p cgwrk 20201028 DE-101 https://d-nb.info/provenance/plan#cgwrk 6\p cgwrk 20201028 DE-101 https://d-nb.info/provenance/plan#cgwrk |
| spellingShingle | Ekeland, Ivar Convexity Methods in Hamiltonian Mechanics Mathematics Differential equations, partial Mathematical optimization Partial Differential Equations Calculus of Variations and Optimal Control; Optimization Theoretical, Mathematical and Computational Physics Game Theory, Economics, Social and Behav. Sciences Mathematik Mechanik (DE-588)4038168-7 gnd Hamiltonsches System (DE-588)4139943-2 gnd Konvexe Analysis (DE-588)4138566-4 gnd Konvexe Menge (DE-588)4165212-5 gnd Hamilton-Funktion (DE-588)4323257-7 gnd Konvexe Funktion (DE-588)4139679-0 gnd Theoretische Mechanik (DE-588)4185100-6 gnd Konvexität (DE-588)4114284-6 gnd Variationsproblem (DE-588)4187419-5 gnd |
| subject_GND | (DE-588)4038168-7 (DE-588)4139943-2 (DE-588)4138566-4 (DE-588)4165212-5 (DE-588)4323257-7 (DE-588)4139679-0 (DE-588)4185100-6 (DE-588)4114284-6 (DE-588)4187419-5 |
| title | Convexity Methods in Hamiltonian Mechanics |
| title_auth | Convexity Methods in Hamiltonian Mechanics |
| title_exact_search | Convexity Methods in Hamiltonian Mechanics |
| title_full | Convexity Methods in Hamiltonian Mechanics by Ivar Ekeland |
| title_fullStr | Convexity Methods in Hamiltonian Mechanics by Ivar Ekeland |
| title_full_unstemmed | Convexity Methods in Hamiltonian Mechanics by Ivar Ekeland |
| title_short | Convexity Methods in Hamiltonian Mechanics |
| title_sort | convexity methods in hamiltonian mechanics |
| topic | Mathematics Differential equations, partial Mathematical optimization Partial Differential Equations Calculus of Variations and Optimal Control; Optimization Theoretical, Mathematical and Computational Physics Game Theory, Economics, Social and Behav. Sciences Mathematik Mechanik (DE-588)4038168-7 gnd Hamiltonsches System (DE-588)4139943-2 gnd Konvexe Analysis (DE-588)4138566-4 gnd Konvexe Menge (DE-588)4165212-5 gnd Hamilton-Funktion (DE-588)4323257-7 gnd Konvexe Funktion (DE-588)4139679-0 gnd Theoretische Mechanik (DE-588)4185100-6 gnd Konvexität (DE-588)4114284-6 gnd Variationsproblem (DE-588)4187419-5 gnd |
| topic_facet | Mathematics Differential equations, partial Mathematical optimization Partial Differential Equations Calculus of Variations and Optimal Control; Optimization Theoretical, Mathematical and Computational Physics Game Theory, Economics, Social and Behav. Sciences Mathematik Mechanik Hamiltonsches System Konvexe Analysis Konvexe Menge Hamilton-Funktion Konvexe Funktion Theoretische Mechanik Konvexität Variationsproblem |
| url | https://doi.org/10.1007/978-3-642-74331-3 |
| work_keys_str_mv | AT ekelandivar convexitymethodsinhamiltonianmechanics |