87 research outputs found
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
Non-existence of an invariant measure for a homogeneous ellipsoid rolling on the plane
It is known that the reduced equations for an axially symmetric homogeneous
ellipsoid that rolls without slipping on the plane possess a smooth invariant
measure. We show that such an invariant measure does not exist in the case when
all of the semi-axes of the ellipsoid have different length.Comment: v2: Minor changes after journal review. This text uses the theory
developed in arXiv:1304.1788 for the specific example of a homogeneous
ellipsoid rolling on the plan
Some remarks about the centre of mass of two particles in spaces of constant curvature
The concept of centre of mass of two particles in 2D spaces of constant
Gaussian curvature is discussed by recalling the notion of "relativistic rule
of lever" introduced by Galperin [Comm. Math. Phys. 154 (1993), 63--84] and
comparing it with two other definitions of centre of mass that arise naturally
on the treatment of the 2-body problem in spaces of constant curvature: firstly
as the collision point of particles that are initially at rest, and secondly as
the centre of rotation of steady rotation solutions. It is shown that if the
particles have distinct masses then these definitions are equivalent only if
the curvature vanishes and instead lead to three different notions of centre of
mass in the general case.Comment: 12 pages, 3 figure
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
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
The inhomogeneous Suslov problem
We consider the Suslov problem of nonholonomic rigid body motion with
inhomogeneous constraints. We show that if the direction along which the Suslov
constraint is enforced is perpendicular to a principal axis of inertia of the
body, then the reduced equations are integrable and, in the generic case,
possess a smooth invariant measure. Interestingly, in this generic case, the
first integral that permits integration is transcendental and the density of
the invariant measure depends on the angular velocities. We also study the
Painlev\'e property of the solutions.Comment: 10 pages, 5 figure
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
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