1,720,967 research outputs found
An elemental overviewof the nonholonomic Noether theorem
Full text linkNoether theorem plays a central role in linking symmetries and first integrals in Lagrangian mechanics. The situation is different in the nonholonomic context, but in the last decades there have been several extensions of Noether theorem to the nonholonomic setting. We provide an overview of this subject which is as elementary as possible
On the Trajectory Generation of the Hydrodynamic Chaplygin Sleigh
Full text linkIn this letter we consider the asymptotic behaviour and the trajectory generation problem for the Chaplygin sleigh interacting with a potential fluid. We investigate which trajectories can be obtained, at least asymptotically as t tents to infinity, by controlling some of the coordinates (shape-control variables) and using the theory of reconstruction. Moreover we support our conclusions via numerical simulations
On the Trajectory Generation of the Hydrodynamic Chaplygin Sleigh
Full text linkIn this letter we consider the asymptotic behaviour and the trajectory generation problem for the Chaplygin sleigh interacting with a potential fluid. We investigate which trajectories can be obtained, at least asymptotically as t tents to infinity, by controlling some of the coordinates (shape-control variables) and using the theory of reconstruction. Moreover we support our conclusions via numerical simulations
Integrable almost--symplectic Hamiltonian systems
Full text linkWe extend the notion of Liouville integrability, which is peculiar to Hamiltonian systems on symplectic manifolds, to Hamiltonian systems on almost-symplectic manifolds, namely, manifolds equipped with a nondegenerate (but not closed) 2-form. The key ingredient is to require that the Hamiltonian vector fields of the integrals of motion in involution (or equivalently, the generators of the invariant tori) are symmetries of the almost-symplectic form. We show that, under this hypothesis, essentially all of the structure of the symplectic case (from quasiperiodicity of motions to an analog of the action-angle coordinates and of the isotropic-coisotropic dual pair structure characteristic of the fibration by the invariant tori) carries over to the almost-symplectic cas
On Some Aspects of the Dynamics of a Ball in a Rotating Surface of Revolution and of the Kasamawashi Art
Full text linkWe study some aspects of the dynamics of the nonholonomic system formed by a
heavy homogeneous ball constrained to roll without sliding on a steadily rotating surface of revolution. First, in the case in which the figure axis of the surface is vertical (and hence the system is SO(3) × SO(2)-symmetric) and the surface has a (nondegenerate) maximum at its vertex, we show the existence of motions asymptotic to the vertex and rule out the possibility of blowup. This is done by passing to the 5-dimensional SO(3)-reduced system.The SO(3)-symmetry persists when the figure axis of the surface is inclined with respect to the vertical — and the system can be viewed as a simple model for the Japanese kasamawashi (turning umbrella) performance art — and in that case we study the (stability of the) equilibria of the 5-dimensional reduced system
The reaction-annihilator distribution and the nonholonomic Noether theorem for lifted actions
No full textWe consider nonholonomic systems with linear, time-independent constraints subject to positional conservative active forces. We identify a distribution on the configuration manifold, that we call the reaction-annihilator distribution R degrees, the fibers of which are the annihilators of the set of all values taken by the reaction forces on the fibers of the constraint distribution. We show that this distribution, which can be effectively computed in specific cases, plays a central role in the study of first integrals linear in the velocities of this class of nonholonomic systems. In particular we prove that, if the Lagrangian is invariant under (the lift of) a group action in the configuration manifold, then an infinitesimal generator of this action has a conserved momentum if and only if it is a section of the distribution R degrees. Since the fibers of R degrees contain those of the constraint distribution, this version of the nonholonomic Noether theorem accounts for more conserved momenta than what was known so far. Some examples are given
Moving energies hide within Noether’s first theorem
Full text linkWe show that the moving energies of some well-known nonholonomic systems
are hidden among the first integrals that can be obtained by applying Noether’s
first Theorem to a suitable Lagrangian
Gauge conservation laws and the momentum equation in nonholonomic mechanics
Full text linkThe gauge mechanism is a generalization of the momentum map which links conservation laws to symmetry groups of nonholonomic systems. This method has been so far employed to interpret conserved quantities as momenta of vector fields which are sections of the constraint distribution. In order to obtain the largest class of conserved quantities of this type, we extend this method to an over-distribution of the constraint distribution, the so-called reaction-annihilator distribution, which encodes the effects that the nonholonomic reaction force has on the conservation laws. We provide examples showing the effectiveness of this generalization. Furthermore, we discuss the Noetherian properties of these conserved quantities, that is, whether and to which extent they depend only on the group, and not on the system. In this context, we introduce a notion of ‘weak Noetherianity’. Finally, we point out that the gauge mechanism is equivalent to the momentum equation (at least for locally free actions), we generalize the momentum equation to the reaction-annihilator distribution, and we introduce a ‘gauge momentum map’ which embodies both methods. For simplicity, we treat only the case of linear constraints, natural Lagrangians, and lifted actions
Remarks on geometric quantum mechanics
Full text linkPursuing the aims of geometric quantum mechanics, it is shown in a geometrical fashion that, at least in finite dimensions, Schroedinger
dynamics enjoys classical complete integrability, and several consequences therefrom are deduced, including a Hannay-type reinterpretation of
Berry's phase and a geometric description of some
aspects of the quantum measurement problem
A Geometric Characterization of Certain First Integrals for Nonholonomic Systems with Symmetries
Full text linkWe study the existence of first integrals in nonholonomic systems with symmetry. First we define the concept of M -cotangent lift of a vector field on a manifold Q in order to unify the works [Balseiro P., Arch. Ration. Mech. Anal. 214 (2014), 453-501, arXiv:1301.1091], [Fosse F., Ramos A., Sansonetto N., Regul. Chaotic Dyn. 12 (2007), 579588], and [Fosse F., Giacobbe A., Sansonetto N., Rep. Math. Phys. 62 (2008), 345-367]. Second, we study gauge symmetries and gauge momenta, in the cases in which there are the symmetries that satisfy the so-called vertical symmetry condition. Under such condition we can predict the number of linearly independent first integrals (that are gauge momenta). We illustrate the theory with two examples
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