## ＜電子ブック＞Convexity Methods in Hamiltonian Mechanics

責任表示 by Ivar Ekeland Ekeland, Ivar SpringerLink (Online service) 1st ed. 1990. English (英語) Springer Berlin Heidelberg Imprint: Springer 1990- Berlin, Heidelberg, Germany シリーズ Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge / A Series of Modern Surveys in Mathematics ; 19 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 perturbati...on theory: there is a small parameter È in the problem, the mass of the perturbing body for instance, and for È = 0 the system becomes completely integrable. One then tries to show that its periodic solutions will subsist for È -# 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.続きを見る I. Linear Hamiltonian Systems1. Floquet Theory and Stability2. Krein Theory and Strong Stability3. Time-Dependence of the Eigenvalues of R (t)4. Index Theory for Positive Definite Systems5. The Iteration Formula6. The Index of a Periodic Solution to a Nonlinear Hamiltonian System7. Examples8. Non-periodic Solutions: The Mean IndexII. Convex Hamiltonian Systems1. Fundamentals of Convex Analysis2. Convex Analysis on Banach Spaces3. Integral Functionals on L?4. The Clarke Duality FormulaIII. Fixed-Period Problems: The Sublinear Case1. Subquadratic Hamiltonians2. An Existence Result3. Autonomous Systems4. Nonautonomous Systems5. Other ProblemsIV. Fixed-Period Problems: The Superlinear Case1. Mountain-Pass Points2. A Preliminary Existence Result3. The Index at Mountain-Pass Points4. Subharmonics5. Autonomous Problems and Potential WellsV. Fixed-Energy Problems1. Existence, Length, Stability2. Multiplicity in the Pinched Case3. Multiplicity in the General Case4. Open Problems.続きを見る http://hdl.handle.net/2324/1000031856 Full text available from SpringerLink ebooks - Mathematics and Statistics (Archive)

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レコードID 3204900 QA370-380 515.353 Differential equations, partial. Mathematical optimization. Mathematics. Partial Differential Equations. Calculus of Variations and Optimal Control; Optimization. Theoretical, Mathematical and Computational Physics. Game Theory, Economics, Social and Behav. Sciences. ssj0001296316 9783642743337(print) 9783540506133(print) 9783642743320(print) 9783642743313 2020.06.27 2020.06.28