Inverse Problems in Ordinary Differential Equations and Applications

ISBN-13: 9783319263373 / Angielski / Twarda / 2016 / 266 str.

Jaume Llibre; Rafael Ramirez

Opis

This book focuses on finding all ordinary differential equations that satisfy a given set of properties. It is dedicated to inverse problems of ordinary differential equations, with the Nambu bracket acting as the central tool to the authors approach. The authors start with a characterization of ordinary differential equations in R DEGREESN which have a given set of M N partial and first integrals, before addressing planar polynomial differential systems with a given set of polynomial partial integrals. They continue solving the 16th Hilbert problem (restricted to algebraic limit cycles) based on generic assumptions, followed by a study of the inverse problem for constrained Lagrange mechanics and Hamiltonian systems, as well as the issue of the integrability of a constrained rigid body. And finally they conclude with an analysis of transpositional relations, a generalization of the Hamiltonian principle, as well as the inverse problem in vakonomic mechanics.
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Recenzje

"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)

Spis treści

Preface.- 1.Differential Equations with Given Partial and First Integrals.- 2.Polynomial Vector Fields with Given Partial and First Integrals.- 3.16th Hilbert Problem for Algebraic Limit Cycles.- 4.Inverse Problem for Constrained Lagrangian Systems.- 5.Inverse Problem for Constrained Hamiltonian Systems.- 6.Integrability of the Constrained Rigid Body.- 7.Inverse Problem in the Vakonomic Mechanics.- Index.- Bibliography.

Blurb

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.