Inverse Problems in Ordinary Differential Equations and Applications

Description

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. Front Matter....Pages i-xii Differential Equations with Given Partial and First Integrals....Pages 1-40 Polynomial Vector Fields with Given Partial and First Integrals....Pages 41-85 Hilbert’s 16th Problem for Algebraic Limit Cycles....Pages 87-116 Inverse Problem for Constrained Lagrangian Systems....Pages 117-152 Inverse Problem for Constrained Hamiltonian Systems....Pages 153-163 Integrability of the Constrained Rigid Body....Pages 165-199 Inverse Problem in Vakonomic Mechanics....Pages 201-251 Back Matter....Pages 253-266