Twisted isotropic realisations of twisted Poisson structures

Motivated by the recent connection between nonholonomic integrable systems and twisted Poisson manifolds made in \cite{balseiro_garcia_naranjo}, this paper investigates the global theory of integrable Hamiltonian systems on almost symplectic manifolds as an initial step to understand Hamiltonian integrability on twisted Poisson (and Dirac) manifolds. Non-commutative integrable Hamiltonian systems on almost symplectic manifolds were first defined in \cite{fasso_sansonetto}, which proved existence of local generalised action-angle coordinates in the spirit of the Liouville-Arnol'd theorem. In analogy with their symplectic counterpart, these systems can be described globally by twisted isotropic realisations of twisted Poisson manifolds, a special case of symplectic realisations of twisted Dirac structures considered in \cite{bursztyn_crainic_weinstein_zhu}. This paper classifies twisted isotropic realisations up to smooth isomorphism and provides a cohomological obstruction to the construction of these objects, generalising the main results of \cite{daz_delz}.Comment: 20 page

Conservation of `moving' energy in nonholonomic systems with affine constraints and integrability of spheres on rotating surfaces

Energy is in general not conserved for mechanical nonholonomic systems with affine constraints. In this article we point out that, nevertheless, in certain cases, there is a modification of the energy that is conserved. Such a function coincides with the energy of the system relative to a different reference frame, in which the constraint is linear. After giving sufficient conditions for this to happen, we point out the role of symmetry in this mechanism. Lastly, we apply these ideas to prove that the motions of a heavy homogeneous solid sphere that rolls inside a convex surface of revolution in uniform rotation about its vertical figure axis, are (at least for certain parameter values and in open regions of the phase space) quasi-periodic on tori of dimension up to three

Hamiltonian monodromy via geometric quantization and theta functions

In this paper, Hamiltonian monodromy is studied from the point of view of geometric quantization abd theta functions, and various differential geometric aspects thereof are dealt with, all related to holonomies of suitable flat connections.Comment: 18 page

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 $n$-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

First Integrals and symmetries of nonholonomic systems

In nonholonomic mechanics, the presence of constraints in the velocities breaks the well-under\-stood link between symmetries and first integrals of holonomic systems, expressed in Noether's Theorem. However there is a known special class of first integrals of nonholonomic systems generated by vector fields tangent to the group orbits, called {\it horizontal gauge momenta}, that suggest that some version of this link should still hold. In this paper we prove that, under certain conditions on the symmetry Lie group, the (nonholonomic) momentum map is conserved along the nonholonomic dynamics, thus extending Noether Theorem to the nonholonomic framework. Our analysis leads to a constructive method, with fundamental consequences to the integrability of some nonholonomic systems as well as their hamiltonization. We apply our results to three paradigmatic examples: the snakeboard, a solid of revolution rolling without sliding on a plane and a heavy homogeneous ball that rolls without sliding inside a convex surface of revolution. In particular, for the snakeboard we show the existence of a new horizontal gauge momentum that reveals new aspects of its integrability

A Geometric Characterization of Certain First Integrals for Nonholonomic Systems with Symmetries

We 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

On the number of weakly Noetherian constants of motion of nonholonomic systems

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Overcoming some drawbacks of Dynamic Movement Primitives

Dynamic Movement Primitives (DMPs) is a framework for learning a point-to-point trajectory from a demonstration. Despite being widely used, DMPs still present some shortcomings that may limit their usage in real robotic applications. Firstly, at the state of the art, mainly Gaussian basis functions have been used to perform function approximation. Secondly, the adaptation of the trajectory generated by the DMP heavily depends on the choice of hyperparameters and the new desired goal position. Lastly, DMPs are a framework for ‘one-shot learning’, meaning that they are constrained to learn from a unique demonstration. In this work, we present and motivate a new set of basis functions to be used in the learning process, showing their ability to accurately approximate functions while having both analytical and numerical advantages w.r.t. Gaussian basis functions. Then, we show how to use the invariance of DMPs w.r.t. affine transformations to make the generalization of the trajectory robust against both the choice of hyperparameters and new goal position, performing both synthetic tests and experiments with real robots to show this increased robustness. Finally, we propose an algorithm to extract a common behavior from multiple observations, validating it both on a synthetic dataset and on a dataset obtained by performing a task on a real robot

On the Dynamics of a Heavy Symmetric Ball that Rolls Without Sliding on a Uniformly Rotating Surface of Revolution

We study the class of nonholonomic mechanical systems formed by a heavy symmetric ball that rolls without sliding on a surface of revolution, which is either at rest or rotates about its (vertical) figure axis with uniform angular velocity Omega. The first studies of these systems go back over a century, but a comprehensive understanding of their dynamics is still missing. The system has an SO(3) x SO(2) symmetry and reduces to four dimensions. We extend in various directions, particularly from the case Omega = 0 to the case Omega not equal 0, a number of previous results and give new results. In particular, we prove that the reduced system is Hamiltonizable even if Omega not equal 0 and, exploiting the recently introduced "moving energy," we give sufficient conditions on the profile of the surface that ensure the periodicity of the reduced dynamics and hence the quasiperiodicity of the unreduced dynamics on tori of dimension up to three. Furthermore, we determine all the equilibria of the reduced system, which are classified in three distinct families, and determine their stability properties. In addition to this, we give a new form of the equations of motion of nonholonomic systems in quasi-velocities which, at variance from the well-known Hamel equations, use any set of quasi-velocities and explicitly contain the reaction forces

On the number of weakly Noetherian constants of motion of nonholonomic systems

We develop a method to give an estimate on the number of functionally independent constants of motion of a nonholonomic system with symmetry which have the so called 'weakly Noetherian' property [22]. We show that this number is bounded from above by the corank of the involutive closure of a certain distribution on the constraint manifold. The effectiveness of the method is illustrated on several examples