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.

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

Inhalt

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.