18 research outputs found
On Integrable Perturbations of Some Nonholonomic Systems
Integrable perturbations of the nonholonomic Suslov, Veselova, Chaplygin and
Heisenberg problems are discussed in the framework of the classical
Bertrand-Darboux method. We study the relations between the Bertrand-Darboux
type equations, well studied in the holonomic case, with their nonholonomic
counterparts and apply the results to the construction of nonholonomic
integrable potentials from the known potentials in the holonomic case
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
Nonholonomic Hamilton-Jacobi Theory via Chaplygin Hamiltonization
We develop Hamilton-Jacobi theory for Chaplygin systems, a certain class of
nonholonomic mechanical systems with symmetries, using a technique called
Hamiltonization, which transforms nonholonomic systems into Hamiltonian
systems. We give a geometric account of the Hamiltonization, identify necessary
and sufficient conditions for Hamiltonization, and apply the conventional
Hamilton-Jacobi theory to the Hamiltonized systems. We show, under a certain
sufficient condition for Hamiltonization, that the solutions to the
Hamilton-Jacobi equation associated with the Hamiltonized system also solve the
nonholonomic Hamilton-Jacobi equation associated with the original Chaplygin
system. The results are illustrated through several examples.Comment: Accepted for publication in Journal of Geometry and Physic
Moving energies as first integrals of nonholonomic systems with affine constraints
In nonholonomic mechanical systems with constraints that are affine (linear
nonhomogeneous) functions of the velocities, the energy is typically not a
first integral. It was shown in [Fass\`o and Sansonetto, JNLS, 26, (2016)]
that, nevertheless, there exist modifications of the energy, called there
moving energies, which under suitable conditions are first integrals. The first
goal of this paper is to study the properties of these functions and the
conditions that lead to their conservation. In particular, we enlarge the class
of moving energies considered in [Fass\`o and Sansonetto, JNLS, 26, (2016)].
The second goal of the paper is to demonstrate the relevance of moving energies
in nonholonomic mechanics. We show that certain first integrals of some well
known systems (the affine Veselova and LR systems), which had been detected on
a case-by-case way, are instances of moving energies. Moreover, we determine
conserved moving energies for a class of affine systems on Lie groups that
include the LR systems, for a heavy convex rigid body that rolls without
slipping on a uniformly rotating plane, and for an -dimensional
generalization of the Chaplygin sphere problem to a uniformly rotating
hyperplane.Comment: 25 pages, 1 figure. Final version prepared according to the
modifications suggested by the referees of Nonlinearit
Structure preserving discretization of time-reparametrized Hamiltonian systems with application to nonholonomic mechanics
We propose a discretization of vector fields that are Hamiltonian up to
multiplication by a positive function on the phase space that may be
interpreted as a time reparametrization. We prove that our method is structure
preserving in the sense that the discrete flow is interpolated to arbitrary
order by the flow of a continuous system possessing the same structure. In
particular, our discretization preserves a smooth measure on the phase space to
arbitrary order. We present applications to a remarkable class of nonholonomic
mechanical systems that allow Hamiltonization. To our best knowledge, these
results provide the first occurrence in the literature of a measure preserving
discretization of measure preserving nonholonomic systems.Comment: 24 pages, 6 figure
Unimodularity and preservation of volumes in nonholonomic mechanics
The equations of motion of a mechanical system subjected to nonholonomic
linear constraints can be formulated in terms of a linear almost Poisson
structure in a vector bundle. We study the existence of invariant measures for
the system in terms of the unimodularity of this structure. In the presence of
symmetries, our approach allows us to give necessary and sufficient conditions
for the existence of an invariant volume, that unify and improve results
existing in the literature. We present an algorithm to study the existence of a
smooth invariant volume for nonholonomic mechanical systems with symmetry and
we apply it to several concrete mechanical examples.Comment: 37 pages, 3 figures; v3 includes several changes to v2 that were done
in accordance to the referee suggestion