37 research outputs found
A Discretization of the Nonholonomic Chaplygin Sphere Problem
The celebrated problem of a non-homogeneous sphere rolling over a horizontal plane was proved to be integrable and was reduced to quadratures by Chaplygin. Applying the formalism of variational integrators (discrete Lagrangian systems) with nonholonomic constraints and introducing suitable discrete constraints, we construct a discretization of the n-dimensional generalization of the Chaplygin sphere problem, which preserves the same first integrals as the continuous model, except the energy. We then study the discretization of the classical 3-dimensional problem for a class of special initial conditions, when an analog of the energy integral does exist and the corresponding map is given by an addition law on elliptic curves. The existence of the invariant measure in this case is also discussed
Discrete Nonholonomic Lagrangian Systems on Lie Groupoids
This paper studies the construction of geometric integrators for nonholonomic
systems. We derive the nonholonomic discrete Euler-Lagrange equations in a
setting which permits to deduce geometric integrators for continuous
nonholonomic systems (reduced or not). The formalism is given in terms of Lie
groupoids, specifying a discrete Lagrangian and a constraint submanifold on it.
Additionally, it is necessary to fix a vector subbundle of the Lie algebroid
associated to the Lie groupoid. We also discuss the existence of nonholonomic
evolution operators in terms of the discrete nonholonomic Legendre
transformations and in terms of adequate decompositions of the prolongation of
the Lie groupoid. The characterization of the reversibility of the evolution
operator and the discrete nonholonomic momentum equation are also considered.
Finally, we illustrate with several classical examples the wide range of
application of the theory (the discrete nonholonomic constrained particle, the
Suslov system, the Chaplygin sleigh, the Veselova system, the rolling ball on a
rotating table and the two wheeled planar mobile robot).Comment: 45 page
A Generalization of Chaplygin's Reducibility Theorem
In this paper we study Chaplygin's Reducibility Theorem and extend its
applicability to nonholonomic systems with symmetry described by the
Hamilton-Poincare-d'Alembert equations in arbitrary degrees of freedom. As
special cases we extract the extension of the Theorem to nonholonomic Chaplygin
systems with nonabelian symmetry groups as well as Euler-Poincare-Suslov
systems in arbitrary degrees of freedom. In the latter case, we also extend the
Hamiltonization Theorem to nonholonomic systems which do not possess an
invariant measure. Lastly, we extend previous work on conditionally variational
systems using the results above. We illustrate the results through various
examples of well-known nonholonomic systems.Comment: 27 pages, 3 figures, submitted to Reg. and Chaotic Dy
On the estimation of stationary level of earthquake catalogs
Abstract:
In this paper we analyzed the stationary level of JMA catalog of magnitude and time intervals between events. It was shown, that these distributions are non-stationary and the time dependence of Gutenberg – Richter law parameter could be represented as a superposition of two quasi-periodical dynamical systems with short and long periods.Note:
Research direction:Mathematical modelling in actual problems of science and technic
Discrete Lagrange-d’Alembert-Poincaré equations for Euler’s disk
Nonholonomic systems are described by the Lagrange-d’Alembert principle. The presence of symmetry leads to a reduced d’Alembert principle and to the Lagrange-d’Alembert-Poincaré equations. First, we briefly recall from previous works how to obtain these reduced equations for the case of a thick disk rolling on a rough surface using a three-dimensional abelian group of symmetries. The main results of the present paper are the calculation of the discrete Lagrange-d’Alembert-Poincaré equations for an Euler’s disk and the numerical simulation of a trajectory and its energy behavior.Fil: Campo, Cédric M.. Universidad Autónoma de Madrid; EspañaFil: Cendra, Hernan. Universidad Nacional del Sur; ArgentinaFil: Diaz, Viviana Alejandra. Universidad Nacional del Sur; ArgentinaFil: de Diego, David Martín. Universidad Autónoma de Madrid; Españ