© 2012

Critical Point Theory for Lagrangian Systems


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

Table of contents

  1. Front Matter
    Pages i-xii
  2. Marco Mazzucchelli
    Pages 1-27
  3. Marco Mazzucchelli
    Pages 29-48
  4. Marco Mazzucchelli
    Pages 49-77
  5. Marco Mazzucchelli
    Pages 79-107
  6. Marco Mazzucchelli
    Pages 109-125
  7. Marco Mazzucchelli
    Pages 127-156
  8. Back Matter
    Pages 157-187

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.


Euler-Lagrange equations Lagrangian dynamics Morse theory periodic orbits

Authors and affiliations

  1. 1.Eberly College of Science, Department of MathematicsPenn State UniversityUniversity ParkUSA

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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)