102,127 research outputs found
The Geodesics for Poincaré’s Half-Plane: a Nonstandard Derivation
No full textConstants of motion in Mechanics are usually inferred from groups of symmetry transformations of the given system, as, for example, a Lagrangian function that is time-invariant implies the conservation of energy. Here we wish to show that useful properties of a mechanical system can sometimes be deduced from a family of Noether-like transformations that are not inspired by any symmetry whatsoever. The sample system we concentrate on is the Lagrangian interpretation of Poincaré’s half plane of hyperbolic geometry, and the properties we will derive in a new way are the shape and the time parameterization of its geodesics
Central potentials with closed cruise orbits
No full textA particle will be said to be in cruise motion if it is nonholonomically constrained to have constant speed. When the particle is placed in a central force field, the resulting mechanical system is known to be integrable. Cruise orbits in a central force field may be closed (periodic in time) or not, depending on the potential and on the speed. Here we give a constructive characterization of all central potentials for which all cruise motions of a given speed are closed. As a consequence, we also give a new proof of the fact that for any such potential the set of speeds for which all cruise motions are closed has always empty interior. (C) 2009 Elsevier Inc. All rights reserved
Complete integrability for Hamiltonian systems with a cone potential
No full textIt is known that, if a point in R^n is driven by a bounded below potential V, whose gradient is always in a closed convex cone which contains no lines, then the velocity has a finite limit as time goes to +infinity. The components of the asymptotic velocity, as functions of the initial data, are trivially constants of motion. We find sufficient conditions for these functions to be C^k (2<k<infinity) first integrals, independent and pairwise in involution. In this way we construct a large class of completely integrable systems. We can deal with very different asymptotic behaviours of the potential and we have persistence of the integrability under any small perturbation of the potential in an arbitrary compact set
Time reversibility and energy conservation for Lagrangian systems with nonlinear nonholonomic constraints
No full textA class of integrable Hamiltonian systems including scattering of particles on the line with repulsive interactions.
No full textThe main purpose of this paper is to introduce a new class of Hamiltonian scattering systems of the cone potential type that can be integrated via the asymptotic velocity. For a large subclass, the asymptotic data of the trajectories define a global canonical diffeomorphism \A that brings the system into the normal form , .
The integrability theory applies for example to a system of particles on the line interacting pairwise through rather general repulsive potentials. The inverse -power potential for arbitrary~ is included, the reduction to normal form being carried out for the exponents~. In particular, the Calogero system is obtained for~. The treatment covers also the nonperiodic Toda lattice.
The cone potentials that we allow can undergo small perturbations in any arbitrary compact set without losing the integrability and the reduction to normal form
Time reversibility and energy conservation for Lagrangian systems with nonlinear nonholonomic constraints
No full textWhen is a nonholonomic Lagrangian system
time-reversible? We prove that a simple sufficient condition is
that (skipping over some minor technicalities) both the Lagrangian
and the set of the triples that
satisfy the constraints are invariant by exchange of into . Another question is: when is energy
conserved in a nonholonomic autonomous Lagrangian system?
A likewise easy sufficient condition is that the set of the
couples satisfying the constraints is a cone
with respect to~ (meaning that if is admissible
then is admissible too for all ).
Time-reversibility and energy conservation are independent
properties, in the sense that neither one implies the other.
Both properties hold at the same time for any autonomous system
with a ``natural'' Lagrangian and with constraints that are
homogenous in
Yagzhev polynomial mappings: on the structure of the Taylor expansion of their local inverse
No full textIt is well known that the Jacobian conjecture follows if it is
proved for the special polynomial mappings f\colon\C^n\to\C^n of
the Yagzhev type: , where is a trilinear
form and . Dru\D{z}kowski and Rusek~\cite{7}
showed that if we take the local inverse of~ at the origin and
expand it into a Taylor series of homogeneous
terms~ of degree~, we find that is a
linear combination of certain ``nested compositions'' of~ with
itself times. If~the Jacobian Conjecture were true,
should be a polynomial mapping of degree~ and the
terms ought to vanish identically for . We
may wonder whether the
reason why vanishes is that {\it each} of the nested
compositions is somehow zero for large~. In this note we show
that this is not at all the case, using a polynomial mapping found
by van den Essen for other purposes
On cubic-linear polynomial mappings
No full textAbstractIn the field of the Jacobian conjecture it is well-known after Drużkowski that from a polynomial ‘cubic-homogeneous’ mapping we can build a higher-dimensional ‘cubic-linear’ mapping and the other way round, so that one of them is invertible if and only if the other one is. We make this point clearer through the concept of ‘pairing’ and apply it to the related conjugability problem: one of the two maps is conjugable if and only if the other one is; moreover, we find simple formulas expressing the inverse or the conjugations of one in terms of the inverse or conjugations of the other. Two nontrivial examples of conjugable cubic-linear mappings are provided as an application
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