Llibre, J.; Ramirez Inostroza, Rafael; Ramirez, V.; Sadovskaia, N. Journal of differential equations Vol. 260, num. 7, p. 5726-5760 DOI: 10.1016/j.jde.2015.12.019 Data de publicació: 2016-04-05 Article en revista
We prove the following two results. First every planar polynomial vector field of degree S with S invariant circles is Darboux integrable without limit cycles. Second a planar polynomial vector field of degree S admits at most S-1 invariant circles which are algebraic limit cycles. In particular we solve the 16th Hilbert problem restricted to algebraic limit cycles given by circles, because a planar polynomial vector field of degree S has at most S-1 algebraic limit cycles given by circles, and this number is reached.
The aim of this paper was to show that the Lagrange-d'Alembert and its equivalent the Gauss and Appel principle are not the only way to deduce the equations of motion of the nonholonomic systems. Instead of them we consider the generalization of the Hamiltonian principle for nonholonomic systems with non-zero transpositional relations. We apply this variational principle, which takes into the account transpositional relations different from the classical ones, and we deduce the equations of motion for the nonholonomic systems with constraints that in general are nonlinear in the velocity. These equations of motion coincide, except perhaps in a zero Lebesgue measure set, with the classical differential equations deduced with the d'Alembert-Lagrange principle. We provide a new point of view on the transpositional relations for the constrained mechanical systems: the virtual variations can produce zero or non-zero transpositional relations. In particular, the independent virtual variations can produce non-zero transpositional relations. For the unconstrained mechanical systems, the virtual variations always produce zero transpositional relations. We conjecture that the existence of the nonlinear constraints in the velocity must be sought outside of the Newtonian mechanics. We illustrate our results with examples.
Llibre, J.; Ramirez Inostroza, Rafael; Sadovskaia, N. Journal of dynamics and differential equations Vol. 26, num. 3, p. 529-581 DOI: 10.1007/s10884-014-9390-1 Data de publicació: 2014-09-01 Article en revista
This paper is on the so called inverse problem of ordinary differential equations, i.e. the problem of determining the differential system satisfying a set of given properties. More precisely we characterize under very general assumptions the ordinary differential equations in which have a given set of either partial integrals, or first integral, or partial and first integrals. Moreover, for such systems we determine the necessary and sufficient conditions for the existence of independent first integrals. We give two relevant applications of the solutions of these inverse problem to constrained Lagrangian and Hamiltonian systems respectively. Additionally we provide the general solution of the inverse problem in dynamics.
The integrability theory for the differential equations, which describe the motion of an unconstrained rigid body around a fixed point is well known. When there are constraints the theory of integrability is incomplete. The main objective of this paper is to analyze the integrability of the equations of motion of a constrained rigid body around a fixed point in a force field with potential U(¿)=U(¿ 1,¿ 2,¿ 3). This motion subject to the constraint <¿,¿>=0 with ¿ is a constant vector is known as the Suslov problem, and when ¿=¿ is the known Veselova problem, here ¿=(¿ 1,¿ 2,¿ 3) is the angular velocity and <¿,¿> is the inner product of R3 .
We provide the following new integrable cases.
(i) The Suslov’s problem is integrable under the assumption that ¿ is an eigenvector of the inertial tensor I and the potential is such that
where I 1,I 2, and I 3 are the principal moments of inertia of the body, µ 1 and µ 2 are solutions of the first-order partial differential equation
(ii) The Veselova problem is integrable for the potential
where ¿ 1 and ¿ 2 are the solutions of the first-order partial differential equation where p=I1I2I3(¿21I1+¿22I2+¿23I3)--------------v .
Also it is integrable when the potential U is a solution of the second-order partial differential equation where t2=I1¿21+I2¿22+I3¿23 and t3=¿21I1+¿22I2+¿23I3 .
Moreover, we show that these integrable cases contain as a particular case the previous known results.
Llibre, J.; Ramirez Inostroza, Rafael; Sadovskaia, N. Journal of dynamics and differential equations Vol. 23, num. 4, p. 885-902 DOI: 10.1007/s10884-011-9219-0 Data de publicació: 2011-07-13 Article en revista
We give an upper bound for the maximum number N of algebraic limit cycles that a planar polynomial vector field of degree n can exhibit if the vector field has exactly k nonsingular irreducible invariant algebraic curves. Additionally we provide sufficient conditions in order that all the algebraic limit cycles are hyperbolic. We also provide lower bounds for N.
For a polynomial planar vector field of degree n ≥ 2 with generic invariant algebraic curves we show that the maximum number of algebraic limit cycles is 1 + (n − 1)(n − 2)/2 when n is even, and (n − 1)(n − 2)/2 when n is odd. Furthermore, these upper bounds are reached.
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.
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.