Inverse Problems in Ordinary Differential Equations and Applications

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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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

“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)

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

  Jaume Llibre

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

  Rafael Ramírez

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