Rafael Ramírez

On the dynamics of nonholonomic systems

In the development of nonholonomic mechanics one can observe recurring confusion over the very equations of motion as well as the deeper questions associated with the geometry and analysis of these equations. First of all, as far as the equations of motion themselves are concerned, the confusion mainly centered on whether or not the equations could be derived from a variational principle in the usual sense. Attempting to dissipate this confusion, in the present paper we deduce a new form of equations of motion which are suitable for both nonholonomic systems with either linear or nonlinear constraints and holonomic systems (A-model). These equations are deduced from the principle of stationary action (or Hamiltonian principle) with nonzero transpositional relations. We show that the well-known equations of motion for nonholonomic and holonomic systems can be deduced from the A-model. For the systems which we call the generalized Vorones-Chaplygin systems we deduce the equations of motion which coincide with the Vorones and Chaplygin equations for the case in which the constraints are linear with respect to the velocity. An additional result is that the transpositional relations are different from zero only for those generalized coordinates whose variations (in accordance with the equations of nonholonomic constraints) are dependent. For the remaining coordinates, the transpositional relations may be zero.

Cartesian approach for constrained mechanical systems with three degrees of freedom

In the history of mechanics, there have been two points of view for studying mechanical systems: The Newtonian and the Cartesian. According the Descartes point of view, the motion of mechanical systems is described by the first-order differential equations in the N dimensional configuration space Q. In this paper we develop the Cartesian approach for mechanical systems with three degrees of freedom and with constraint which are linear with respect to velocity. The obtained results we apply to discuss the integrability of the geodesic flows on the surface in the three dimensional Euclidian space and to analyze the integrability of a heavy rigid body in the Suslov and the Veselov cases.

On the construction of dynamic systems from given integrals

In this communication we propose a new approach for studying a particular type of inverse problems in mechanics related to the construction of a force field from given integrals.An extension of the Danielli problem is obtained. The given results are applied to the Suslov problem, and illustrated in specific examples.

Polynomial Vector Fields with Given Partial and First Integrals

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.

Inverse Problem in Vakonomic Mechanics

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.

Inverse Problem for Constrained Lagrangian Systems

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.

Hilbert’s 16th Problem for Algebraic Limit Cycles

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.

Differential Equations with Given Partial and First Integrals

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.

Inverse Problem for Constrained Hamiltonian Systems

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]).