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Critical Point Theory for Lagrangian Systems

  • Book
  • © 2012

Overview

Authors:
  1. Marco Mazzucchelli

    1. Eberly College of Science, Department of Mathematics, Penn State University, University Park, USA

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  • Collects, in a rigorous and consistent style, many important results that are sparse in the literature
  • Exposition is self-contained
  • Arguments are presented in an elementary way in order to be accessible to the non-specialists
  • Includes supplementary material: sn.pub/extras

Part of the book series: Progress in Mathematics (PM, volume 293)

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Table of contents (6 chapters)

  1. Front Matter

    Pages i-xii
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  2. Lagrangian and Hamiltonian Systems

    • Marco Mazzucchelli
    Pages 1-27

  3. The Morse Indices in Lagrangian Dynamics

    • Marco Mazzucchelli
    Pages 29-48

  4. Functional Setting for the Lagrangian Action

    • Marco Mazzucchelli
    Pages 49-77

  5. Discretizations

    • Marco Mazzucchelli
    Pages 79-107

  6. Local Homology and Hilbert Subspaces

    • Marco Mazzucchelli
    Pages 109-125

  7. Periodic Orbits of Tonelli Lagrangian Systems

    • Marco Mazzucchelli
    Pages 127-156

  8. Back Matter

    Pages 157-187
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Keywords

About this book

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 investigate existence 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.

Reviews

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)

Authors and Affiliations

  • Eberly College of Science, Department of Mathematics, Penn State University, University Park, USA

    Marco Mazzucchelli

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