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## Über dieses Buch

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.

## Inhaltsverzeichnis

### Chapter 1. Differential Equations with Given Partial and First Integrals

Abstract
In this chapter we present two different kind of results. First, under very general assumptions we characterize the ordinary differential equations in $$\mathbb{R}^N$$ which have a given set of either $$M \leq N,\,or\,M > N$$ partial integrals, or $$M < N$$ first integrals, or $$M \leq N$$ partial and first integrals. Second, in $$\mathbb{R}^N$$ we provide some results on integrability, in the sense that the characterized differential equations admit N – 1 independent first integrals.
Jaume Llibre, Rafael Ramírez

### Chapter 2. Polynomial Vector Fields with Given Partial and First Integrals

Abstract
The solutions of the inverse problem in ordinary differential equations have a very high degree of arbitrariness because of the unknown functions involved. To reduce this arbitrariness we need additional conditions. In this chapter we are mainly interested in the planar polynomial differential systems which have a given set of invariant algebraic curves.
Jaume Llibre, Rafael Ramírez

### Chapter 3. Hilbert’s 16th Problem for Algebraic Limit Cycles

Abstract
In this chapter we state Hilbert’s 16th problem restricted to algebraic limit cycles. Namely, consider the set Σ n of all real polynomial vector fields $$\chi = \left( {P,\,Q} \right)$$ of degree n having real irreducible $$\left( {{\rm on}\, \mathbb{R}\left[ {x,\,y} \right]} \right)$$ invariant algebraic curves.
Jaume Llibre, Rafael Ramírez

### Chapter 4. Inverse Problem for Constrained Lagrangian Systems

Abstract
The aim of this chapter is to provide a solution of the inverse problem of the constrained Lagrangian mechanics which can be stated as follows: Determine for a given natural mechanical system with N degrees of freedom the most general field of forces depending only on the positions and satisfying a given set of constraints with are linear in the velocities. This statement of the inverse problem for constrained Lagrangian systems is new.
Jaume Llibre, Rafael Ramírez

### Chapter 5. Inverse Problem for Constrained Hamiltonian Systems

Abstract
Constrained Hamiltonian systems arise in many fields, for instance in multi-body dynamics or in molecular dynamics. The theory of such systems goes back to by P.A.M. Dirac (see for instance [44]).
Jaume Llibre, Rafael Ramírez

### Chapter 6. Integrability of the Constrained Rigid Body

Abstract
The integration theory of the differential equations which describe the motion of a mechanical system with constraints (constrained mechanical system) is not so complete as for mechanical systems without constraints (unconstrained mechanical systems). This can be due to several reasons. One of them is that the equations of motion of a constrained mechanical system in general have no invariant measure, in contrast to the unconstrained case, see for instance [91].
Jaume Llibre, Rafael Ramírez

### Chapter 7. Inverse Problem in Vakonomic Mechanics

Abstract
The mechanical systems free of constraints are called Lagrangian systems or holonomic systems. The mechanical systems with integrable constraints are called holonomic constrained mechanical systems. Finally, the mechanical systems with non-integrable constraints are usually called nonholonomic mechanical systems, or nonholonomic constrained mechanical systems.
Jaume Llibre, Rafael Ramírez

### Backmatter

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