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Birkhäuser

Inverse Problems in Ordinary Differential Equations and Applications

  • Book
  • © 2016

Overview

Authors:
  1. Jaume Llibre

    1. Departament de Matemàtiques, Universitat Autònoma de Barcelona, Barcelona, Spain

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  2. Rafael Ramírez

    1. Departament d'Enginyeria Informàtica, Universitat Rovira i Virgili, Tarragona, Catalonia, Spain

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  • Solves the 16th Hilbert problem (restricted to algebraic limit cycles) based on generic assumptions
  • Presents a detailed analysis of transpositional relations, a generalization of the Hamiltonian principle
  • Features the Nambu bracket as central tool in the authors' approach on solving inverse problems in ODEs

Part of the book series: Progress in Mathematics (PM, volume 313)

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Table of contents (7 chapters)

  1. Front Matter

    Pages i-xii
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  2. Differential Equations with Given Partial and First Integrals

    • Jaume Llibre, Rafael Ramírez
    Pages 1-40

  3. Polynomial Vector Fields with Given Partial and First Integrals

    • Jaume Llibre, Rafael Ramírez
    Pages 41-85

  4. Hilbert’s 16th Problem for Algebraic Limit Cycles

    • Jaume Llibre, Rafael Ramírez
    Pages 87-116

  5. Inverse Problem for Constrained Lagrangian Systems

    • Jaume Llibre, Rafael Ramírez
    Pages 117-152

  6. Inverse Problem for Constrained Hamiltonian Systems

    • Jaume Llibre, Rafael Ramírez
    Pages 153-163

  7. Integrability of the Constrained Rigid Body

    • Jaume Llibre, Rafael Ramírez
    Pages 165-199

  8. Inverse Problem in Vakonomic Mechanics

    • Jaume Llibre, Rafael Ramírez
    Pages 201-251

  9. Back Matter

    Pages 253-266
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Keywords

About this book

This book is dedicated to study the inverse problem of ordinary differential equations, that is it focuses in finding all ordinary differential equations that satisfy a given set of properties. The Nambu bracket is the central tool in developing this approach. The authors start characterizing the ordinary differential equations in R^N which have a given set of partial integrals or first integrals. The results obtained are applied first to planar polynomial differential systems with a given set of such integrals, second to solve the 16th Hilbert problem restricted to generic algebraic limit cycles, third for solving the inverse problem for constrained Lagrangian and Hamiltonian mechanical systems, fourth for studying the integrability of a constrained rigid body. Finally the authors conclude with an analysis on nonholonomic mechanics, a generalization of the Hamiltonian principle, and the statement an solution of the inverse problem in vakonomic mechanics.

Reviews

“The book presents a new approach to … inverse problems, where the authors mainly use as an essential tool the Nambu bracket. They deduce new properties of this bracket, which plays a fundamental role in the proof of all the results and in their applications throughout the book. … The book is well written and contains new and valuable results in the development of the inverse problem in ordinary differential equations and its applications.” (Leonardo Colombo, Mathematical Reviews, January, 2017)

Authors and Affiliations

  • Departament de Matemàtiques, Universitat Autònoma de Barcelona, Barcelona, Spain

    Jaume Llibre

  • Departament d'Enginyeria Informàtica, Universitat Rovira i Virgili, Tarragona, Catalonia, Spain

    Rafael Ramírez

Bibliographic Information

  • Book Title: Inverse Problems in Ordinary Differential Equations and Applications

  • Authors: Jaume Llibre, Rafael Ramírez

  • Series Title: Progress in Mathematics

  • DOI: https://doi.org/10.1007/978-3-319-26339-7

  • Publisher: Birkhäuser Cham

  • eBook Packages: Mathematics and Statistics, Mathematics and Statistics (R0)

  • Copyright Information: Springer International Publishing Switzerland 2016

  • Hardcover ISBN: 978-3-319-26337-3Published: 22 March 2016

  • Softcover ISBN: 978-3-319-79935-3Published: 24 April 2018

  • eBook ISBN: 978-3-319-26339-7Published: 09 March 2016

  • Series ISSN: 0743-1643

  • Series E-ISSN: 2296-505X

  • Edition Number: 1

  • Number of Pages: XII, 266

  • Number of Illustrations: 1 b/w illustrations, 8 illustrations in colour

  • Topics: Ordinary Differential Equations

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