33 research outputs found
Gauge Transformations, Twisted Poisson Brackets and Hamiltonization of Nonholonomic Systems
In this paper we study the problem of Hamiltonization of nonholonomic systems
from a geometric point of view. We use gauge transformations by 2-forms (in the
sense of Severa and Weinstein [29]) to construct different almost Poisson
structures describing the same nonholonomic system. In the presence of
symmetries, we observe that these almost Poisson structures, although gauge
related, may have fundamentally different properties after reduction, and that
brackets that Hamiltonize the problem may be found within this family. We
illustrate this framework with the example of rigid bodies with generalized
rolling constraints, including the Chaplygin sphere rolling problem. We also
see how twisted Poisson brackets appear naturally in nonholonomic mechanics
through these examples
Hamiltonization and Integrability of the Chaplygin Sphere in R^n
The paper studies a natural -dimensional generalization of the classical
nonholonomic Chaplygin sphere problem. We prove that for a specific choice of
the inertia operator, the restriction of the generalized problem onto zero
value of the SO(n-1)-momentum mapping becomes an integrable Hamiltonian system
after an appropriate time reparametrization.Comment: 22 pages, Sections 1 and 5 are rewritten, to appear in Journal of
Nonlinear Scienc
Quasi-Chaplygin Systems and Nonholonimic Rigid Body Dynamics
We show that the Suslov nonholonomic rigid body problem can be regarded
almost everywhere as a generalized Chaplygin system. Furthermore, this provides
a new example of a multidimensional nonholonomic system which can be reduced to
a Hamiltonian form by means of Chaplygin reducing multiplier. Since we deal
with Chaplygin systems in the local sense, the invariant manifolds of the
integrable examples are not necessary tori.Comment: minor changes, to appear in Letters in Mathematical Physic
Hamiltonisation, measure preservation and first integrals of the multi-dimensional rubber Routh sphere
We consider the multi-dimensional generalisation of the problem of a sphere,
with axi-symmetric mass distribution, that rolls without slipping or spinning
over a plane. Using recent results from Garc\'ia-Naranjo (arXiv: 1805:06393)
and Garc\'ia-Naranjo and Marrero (arXiv: 1812.01422), we show that the reduced
equations of motion possess an invariant measure and may be represented in
Hamiltonian form by Chaplygin's reducing multiplier method. We also prove a
general result on the existence of first integrals for certain Hamiltonisable
Chaplygin systems with internal symmetries that is used to determine conserved
quantities of the problem.Comment: 23 pages, 1 figure. Submitted to the special issue of Theor. Appl.
Mech. in honour of Chaplygin's 150th anniversar
Integrable Euler top and nonholonomic Chaplygin ball
We discuss the Poisson structures, Lax matrices, -matrices, bi-hamiltonian
structures, the variables of separation and other attributes of the modern
theory of dynamical systems in application to the integrable Euler top and to
the nonholonomic Chaplygin ball.Comment: 25 pages, LaTeX with AMS fonts, final versio