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Critical Point Theory for Lagrangian Systems 2012 ed. [Kõva köide]

  • Formaat: Hardback, 188 pages, kõrgus x laius: 235x155 mm, kaal: 471 g, XII, 188 p., 1 Hardback
  • Sari: Progress in Mathematics 293
  • Ilmumisaeg: 11-Nov-2011
  • Kirjastus: Birkhauser Verlag AG
  • ISBN-10: 3034801629
  • ISBN-13: 9783034801621
Teised raamatud teemal:
Critical Point Theory for Lagrangian Systems 2012 ed.Suurem pilt
  • Formaat: Hardback, 188 pages, kõrgus x laius: 235x155 mm, kaal: 471 g, XII, 188 p., 1 Hardback
  • Sari: Progress in Mathematics 293
  • Ilmumisaeg: 11-Nov-2011
  • Kirjastus: Birkhauser Verlag AG
  • ISBN-10: 3034801629
  • ISBN-13: 9783034801621
Teised raamatud teemal:
Püsilink: https://www.kriso.ee/db/9783034801621.html
Märksõnad:
Here is a modern account of the application of critical point theory, specifically Morse theory, to Lagrangian dynamics, with particular emphasis on the existence and multiplicity of periodic orbits of non-autonomous and time-periodic systems.

Lagrangian systems constitute a very important and old class in dynamics. Their origin dates back to the end of the eighteenth century, with Joseph-Louis Lagrange’s reformulation of classical mechanics. The main feature of Lagrangian dynamics is its variational flavor: orbits are extremal points of an action functional. The development of critical point theory in the twentieth century provided a powerful machinery to investigateexistence and multiplicity questions for orbits of Lagrangian systems. This monograph gives a modern account of the application of critical point theory, and more specifically Morse theory, to Lagrangian dynamics, with particular emphasis toward existence and multiplicity of periodic orbits of non-autonomous and time-periodic systems.

Arvustused

From the reviews:

This monograph concerns the use of critical point theory tools in connection with questions of existence and multiplicity of periodic solutions of Lagrangian systems. The monograph contains several proofs and seems to be especially suitable for researchers and advanced graduate students interested in applications of critical point theory to boundary value problems for Lagrangian systems. (Maria Letizia Bertotti, Mathematical Reviews, September, 2013)

The results of critical point theory provide powerful techniques to investigate and study aspects of Lagrangian systems such as existence, multiplicity or uniqueness of solutions of the Euler-Lagrange equations with prescribed boundary conditions. A bibliography with 88 entries, a list of symbols distributed on each chapter, and a subject index complete the work. The book is self-contained and rigorously presented. Various aspects of it should be of interest to graduate students and researchers in this dynamic field of mathematics. (Dorin Andrica, Zentralblatt MATH, Vol. 1246, 2012)

Preface ix
1 Lagrangian and Hamiltonian Systems
1.1 The formalism of classical mechanics
1(6)
1.2 Tonelli systems
7(4)
1.3 Action minimizers
11(18)
2 The Morse Indices in Lagrangian Dynamics
2.1 The Morse index and nullity
29(6)
2.2 Bott's iteration theory
35(7)
2.3 A symplectic excursion: the Maslov index
42(8)
3 Functional Setting for the Lagrangian Action
3.1 Hilbert manifold structures for path spaces
50(4)
3.2 Topological properties of the free loop space
54(5)
3.3 Convex quadratic-growth Lagrangians
59(4)
3.4 Regularity of the action functional
63(7)
3.5 Critical points of the action functional
70(7)
3.6 The mean action functional in higher periods
77(3)
4 Discretizations
4.1 Uniqueness of the action minimizers
80(6)
4.2 The broken Euler-Lagrange loop spaces
86(3)
4.3 The discrete action functional
89(2)
4.4 Critical points of the discrete action
91(8)
4.5 Homotopic approximation of the action sublevels
99(4)
4.6 Multiplicity of periodic orbits with prescribed period
103(3)
4.7 Discretizations in higher period
106(4)
5 Local Homology and Hilbert Subspaces
5.1 The abstract result
110(2)
5.2 The generalized Morse lemma revisited
112(5)
5.3 Naturality of the Morse lemma
117(2)
5.4 Local homology
119(4)
5.5 Application to the action functional
123(5)
6 Periodic Orbits of Tonelli Lagrangian Systems
6.1 Convex quadratic modifications
128(5)
6.2 Multiplicity of periodic orbits with prescribed period
133(2)
6.3 Discrete Tonelli action
135(3)
6.4 Homological vanishing by iteration
138(13)
6.5 The Conley conjecture
151(6)
Appendix: An Overview of Morse Theory
157(16)
A.1 Preliminaries
158(1)
A.2 The generalized Morse Lemma
159(1)
A.3 Deformation of sublevels
160(5)
A.4 Passing a critical level
165(3)
A.5 Local homology and Gromoll-Meyer pairs
168(2)
A.6 Minimax
170(3)
Bibliography 173(6)
List of Symbols 179(6)
Index 185