© 2016

Inverse Problems in Ordinary Differential Equations and Applications


  • 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 Progress in Mathematics book series (PM, volume 313)

Table of contents

  1. Front Matter
    Pages i-xii
  2. Jaume Llibre, Rafael Ramírez
    Pages 1-40
  3. Jaume Llibre, Rafael Ramírez
    Pages 41-85
  4. Jaume Llibre, Rafael Ramírez
    Pages 87-116
  5. Jaume Llibre, Rafael Ramírez
    Pages 117-152
  6. Jaume Llibre, Rafael Ramírez
    Pages 153-163
  7. Jaume Llibre, Rafael Ramírez
    Pages 165-199
  8. Jaume Llibre, Rafael Ramírez
    Pages 201-251
  9. Back Matter
    Pages 253-266

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.


16th Hilbert problem Nambu bracket inverse problems ordinary differential equations planar polynomial differential systems vakonomic mechanics

Authors and affiliations

  1. 1.Departament de MatemàtiquesUniversitat Autònoma de BarcelonaBarcelonaSpain
  2. 2.Departament d'Enginyeria InformàticaUniversitat Rovira i VirgiliTarragona, CataloniaSpain

Bibliographic information

  • Book Title Inverse Problems in Ordinary Differential Equations and Applications
  • Authors Jaume Llibre
    Rafael Ramírez
  • Series Title Progress in Mathematics
  • Series Abbreviated Title Progress in Mathematics(Birkhäuser)
  • DOI
  • Copyright Information Springer International Publishing Switzerland 2016
  • Publisher Name Birkhäuser, Cham
  • eBook Packages Mathematics and Statistics Mathematics and Statistics (R0)
  • Hardcover ISBN 978-3-319-26337-3
  • Softcover ISBN 978-3-319-79935-3
  • eBook ISBN 978-3-319-26339-7
  • 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
  • Buy this book on publisher's site


“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)