Francesco Fassò's research while affiliated with University of Padova and other places
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Publications (52)
In this paper, we give a new characterization of the cut locus of a point on a compact Riemannian manifold as the zero set of the optimal transport density solution of the Monge–Kantorovich equations, a PDE formulation of the optimal transport problem with cost equal to the geodesic distance. Combining this result with an optimal transport numerica...
![Generatrices Γ\documentclass[12pt]{minimal} \usepackage{amsmath}...](publication/363475769/figure/fig1/AS:11431281098996152@1669171839217/Generatrices-Gdocumentclass12ptminimal-usepackageamsmath-usepackagewasysym_Q320.jpg)
![Graphs of Ω~(r,vθ)\documentclass[12pt]{minimal} \usepackage{amsmath}...](publication/363475769/figure/fig5/AS:11431281098991480@1669171839389/Graphs-of-Or-vthdocumentclass12ptminimal-usepackageamsmath-usepackagewasysym_Q320.jpg)
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 $$\Omega $$ Ω . The first studies of these systems go back over a century, but a comprehensive understanding o...
We 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 $\textrm{SO(3)}\times\textrm{SO(2)}$-symmetric) and the surface has a (n...
We 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 \(\textrm{SO(3)}\times\textrm{SO(2)}\)-symmetric) and the surface has a...
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. The first studies of these systems go back over a century, but a comprehensive understanding of their dynamic...
In this paper, we give a new characterization of the cut locus of a point on a compact Riemannian manifold as the zero set of the optimal transport density solution of the Monge-Kantorovich equations, a PDE formulation of the optimal transport problem with cost equal to the geodesic distance. Combining this result with an optimal transport numerica...
![The image and critical rays Lij\documentclass[12pt]{minimal}...](profile/James-Montaldi/publication/324745170/figure/fig2/AS:960073736126514@1605910824779/The-image-and-critical-rays-Lijdocumentclass12ptminimal-usepackageamsmath_Q320.jpg)
![The reduced space R\documentclass[12pt]{minimal} \usepackage{amsmath}...](profile/James-Montaldi/publication/324745170/figure/fig3/AS:960073740341263@1605910825485/The-reduced-space-Rdocumentclass12ptminimal-usepackageamsmath_Q320.jpg)
We consider the the n-dimensional generalisation of the nonholonomic Veselova problem. We derive the reduced equations of motion in terms of the mass tensor of the body and determine some general properties of the dynamics. In particular we give a closed formula for the invariant measure, we indicate the existence of steady rotation solutions, and...
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 a...
The three-dimensional champagne bottle system contains no mondromy, despite being entirely composed of invariant two-dimensional champagne bottle systems, each of which posesses nontrivial monodromy. We explain where the monodromy went in the three-dimensional system, or perhaps, where it did come from in the two-dimensional system, by regarding th...
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...
We return to the Keplerian or n-shell approximation to the hydrogen atom in the presence of weak static electric and magnetic fields. At the classical level, this is a Hamiltonian system with the phase space S
2 × S
2. Its principal order Hamiltonian H
0 was known already to Pauli in 1926. H
0 defines an isochronous system with a linear flow on S
2...
At variance from the cases of relative equilibria and relative periodic
orbits of dynamical systems with symmetry, the dynamics in relative
quasi-periodic tori (namely, subsets of the phase space that project to an
invariant torus of the reduced system on which the flow is quasi-periodic) is
not yet completely understood. Even in the simplest situa...
We characterize the conditions for the conservation of the energy and of the
components of the momentum maps of lifted actions, and of their `gauge-like'
generalizations, in time-independent nonholonomic mechanical systems with
affine constraints. These conditions involve geometrical and mechanical
properties of the system, and are codified in the...
By examining the linkage between conservation laws and symmetry, we explain why it appears there should not be an analogue of a complete integral for the Hamilton-Jacobi equation for integrable 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
(see Fassò et al Rep. Math. Phys. 62 (2008), 345-367). We show that this number is bounded from above by the corank of the involutive closure of a certain...
The notion of gauge momenta is a generalization of the
momentum map which is relevant for nonholonomic systems with symmetry. Weakly Noetherian functions are functions which are constants of motion of all `natural' nonholonomic systems with a given kinetic energy and any G--invariant potential energy. We show that, when the action of the symmetry g...
Noether 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.
This article investigates the computation of the exact free rigid body motion as a component of splitting methods for rigid bodies subject to external forces. We review various matrix and quaternion representations of the solution of the free rigid body equation which involve Jacobi ellipic functions and elliptic integrals and are amenable to numer...
We give a standard model for the flat affine geometry defined by the local action variables of a completely integrable system.
We are primarily interested in the affine structure in the neighborhood of a critical value with nontrivial monodromy.
We 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°, the fibers
of which are the annihilators of the set of all values taken by the reaction forces on the fibers of...
We 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 involutio...
Many and important integrable Hamiltonian systems are ‘superintegrable’, in the sense that there is an open subset of their 2d-dimensional phase space in which all motions are linear on tori of dimension nd. A thorough comprehension of these systems requires a description which goes beyond the standard notion of Liouville–Arnold integrability, that...
It has been observed recently that certain (reduced) nonholonomic systems are Hamiltonian with respect to a rank-two Poisson structure. We link the existence of these structures to a dynamical property of the (reduced) system: its periodicity, with positive period depending continuously on the initial data. Moreover, we show that there are in fact...
We study the long term stability of the proper rotations of the perturbed Euler rigid body, in the framework of Nekhoroshev theory. For simplicity we treat here in detail only the kinetically symmetric case (the potential needs not to be symmetric), but we indicate how to extend the results to the triaxial case. We show that the proper rotations ar...
Directional Quasi--Convexity (DQC) is a su#cient condition for Nekhoroshev stability, that is, stability for finite but very long times, of elliptic equilibria of Hamiltonian systems. The numerical detection of DQC is elementary for system with three degrees of freedom. In this article, we propose a recursive algorithm to test DQC in any number n 4...
We compare several different second-order splitting algorithms for the asymmetric rigid body, with the aim of determining which one produces the smallest energy error for a given rigid body, namely, for given moments of inertia. The investigation is based on the analysis of the dominant term of the modified Hamiltonian and indicates that different...
We study the geometry of the fibration in invariant tori of a Hamiltonian system which is integrable in Bogoyavlenskij’s “broad sense”—a generalization of the standard cases of Liouville and non-commutative integrability. We show that the structure of such a fibration generalizes that of the standard cases. Firstly, the base manifold has a Poisson...
We obtain a global version of the Hamiltonian KAM theorem for in-variant Lagrangean tori by glueing together local KAM conjugacies with help of a partition of unity. In this way we find a global Whit-ney smooth conjugacy between a nearly-integrable system and an inte-grable one. This leads to preservation of geometry, which allows us to define all...
We numerically investigate the dynamics of a symmetric rigid body with a fixed point in a small analytic external potential (equivalently, a fast rotating body in a given external field) in the light of previous theoretical investigations based on Nekhoroshev theory. Special attention is posed on “resonant” motions, for which the tip of the unit ve...
We study the ellipticity and the ``Nekhoroshev stability'' (stability properties for finite, but very long, time scales) of the Riemann ellipsoids. We provide numerical evidence that the regions of ellipticity of the ellipsoids of types II and III are larger than those found by Chandrasekhar in the 60's and that all Riemann ellipsoids, except a fin...
Hamiltonian perturbation theory explains how symplectic integrators work and, in particular, why they can be used to measure extremely small energy exchanges between different degrees of freedom in molecular collision problems. Conversely, numerical experiments based on symplectic integrators permit a detailed understanding of the dynamics of nearl...
In this paper we study the dynamics of a fast rotating symmetric rigid body in an arbitrary analytic potential, by means of the techniques of Hamiltonian perturbation theory, specifically Nekhoroshev's theory. For a rigid body with a fixed point, this approach allows us to get an accurate description of the motion (with rigorous estimates) on time...
The Lagrangian equilateral points L
4 and L
5 of the restricted circular three-body problem are elliptic for all values of the reduced mass μ, below Routh’s critical mass μ
R
≈ .0385. In the spatial case, because of the possibility of Arnold diffusion, KAM theory does not provide Lyapunov-stability. Nevertheless, one can consider the so-called ‘Nek...
We continue the analysis which began in part 1 of this paper of the long-time behaviour of the fast rotations of a rigid body in an external analytic force field. Specifically, we consider the motions of a symmetric rigid body whose angular velocity is nearly parallel to the symmetry axis of the ellipsoid of inertia, which were excluded from the pr...
The Lagrangian equilateral points L 4 and L 5 of the restricted circular three-body problem are elliptic for all values of the reduced mass μ below Routh’s critical mass μ R ≈ .0385. In the spatial case, because of the possibility of Arnold diffusion, KAM theory does not provide Lyapunov-stability. Nevertheless, one can consider the so-called ‘Nekh...
Consider a Hamiltonian system with $d$ degrees of freedom whose motions are all linear on tori of some fixed dimension $n\led$ is such a system necessarily completely (or else non-commutatively) integrable? We show that the answer is affirmative under quite broad conditions, but not always, and we provide counterexamples
We prove a conjecture by N.N. Nekhoroshev about the long-time stability of elliptic equilibria of Hamiltonian systems, without any Diophantine condition on the frequencies. Higher order terms of the Hamiltonian are used to provide convexity. The singularity of the action-angle coordinates at the origin is overcome by working in cartesian coordinate...
It is known that any integrable, possibly degenerate, Hamiltonian system is Hamiltonian relative to many different symplectic structures; under certain hypotheses, the 'semi-local' structure of these symplectic forms, written in local coordinates of action-angle type, is also known. The purpose of this paper is to characterize from the point of vie...
We study the accuracy of the conservation of adiabatic invariants in a model of n weakly coupled rotators. Most attention is devoted to n = 2 and frequency ω = (ω1,ω2), with ω2ω1 quadratic irrational. We apply a heuristic approximation scheme, going back to Jeans and to Landau and Teller, and perform a very accurate numerical check of the result, o...
We study the global structure of the fibration by the invariant two-dimensional tori of the Euler-Poinsot top — the rigid body with a fixed point and no torques. We base our analysis on the notion of bifibration (or dual pair) which, as results from the approach based on the so-called non-commutative integrability, provides a thorough description o...
This paper deals with Hamiltonian perturbation theory for systems which, like Euler-Poinsot (the rigid body with a fixed point and no torques), are degenerate and do not possess a global system of action-angle coordinates. It turns out that the usual methods of perturbation theory, which are essentially local being based on the construction of norm...
Hamiltonian perturbation theory is usually formulated with reference to systems defined in a product space B × T
m
endowed with a system of action-angle coordinates I ∈ B,φ ∈ T
m
, where B is an open set in R
m
. This is essentially a ‘local’ formulation since the phase space of an integrable Hamiltonian system can easily fail to have such a produc...
Using simple known methods and results of classical perturbation theory, especially those due to Nekhoroshev and Neishtadt, we study the energy exchanges between the rotational and the translational degrees of freedom in a particular model representing the planar motion of a rigid body in a bounded analytic potential. We prove that, if the angular...
We consider a rigorous Hamiltonian perturbation theory based on the transformation of the vector field of the system, realized by the Lie method. Such a perturbative technique presents some advantages over the standard one, which uses the transformation of the Hamilton functions. Indeed, the present method is simple, and furnishes quite detailed in...
In this paper we discuss the relation between the structures of the series expansion for the Dragt and Finn composition of Lie transforms and for a transformation introduced by Giorgilli and Galgani. A recursive algorithm is presented which is used to generate the series expansion for the composition of Lie transforms. This algorithm strongly resem...
In this paper we exhibit a rigorous perturbation theory for nearly integrable Hamiltonian systems, based on the composition of Lie Transforms. Precisely, we first study the algorithm for the composition of Lie transforms, and provide rigorous estimates for the convergence radius and the truncation errors of the series; then we use our estimates for...
Citations
... It turns out that the proposed OTPbased approach on surfaces is a general method for the computation of the distance function. Indeed, the method has been applied in [27] for the identification of the cut locus of a triangulated surface with respect of a point, i.e., the set of points where the distance function becomes non differentiable, or, equivalently, the minimizing geodesics are not unique. Moreover, our approach can be extended to the approximation of medial axes and Voronoi diagrams of general surfaces embedded in R 3 , quantities that are of great interest in computational geometry [34]. ...
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... This is shown in [23,56] using the same methods of analysis (in particular, reduction by symmetries). In [31,35] the rolling motion of a sphere on an arbitrary rotating surface of revolution is investigated, and the authors of [16,39] examine integrable cases of this problem: the rolling motions of a sphere on the rotating surface of a cone and a circular cylinder. In [32,33] a new interesting problem concerning the rolling of spheres between two spherical surfaces is addressed and cases of its integrability are found, and in [18] the integrability of the related problem of the motion of a rigid body in a spherical suspension is shown [36]. ...
![[object Object]](https://i1.rgstatic.net/ii/profile.image/540644483362817-1505911097090_Q64/Nicola-Sansonetto.jpg)
... This is shown in [23,56] using the same methods of analysis (in particular, reduction by symmetries). In [31,35] the rolling motion of a sphere on an arbitrary rotating surface of revolution is investigated, and the authors of [16,39] examine integrable cases of this problem: the rolling motions of a sphere on the rotating surface of a cone and a circular cylinder. In [32,33] a new interesting problem concerning the rolling of spheres between two spherical surfaces is addressed and cases of its integrability are found, and in [18] the integrability of the related problem of the motion of a rigid body in a spherical suspension is shown [36]. ...
... In this section, we show numerical simulations of the behavior of some of the brackets computed in Section 4, illustrating the trajectory obtained by iterating the same control loop several times. As discussed in [14], the iteration of a control loop leads to one of two alternative behaviors: either the trajectory is drifting along a certain direction, or it is quasi periodic, remaining in a compact set. More precisely, each zero-mean control loop (thus inducing a gait) produces a displacement, called geometric phase [21,29], in the variable g. ...
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... The existence of an invariant measure for nonholomic problems is well studied [see Fedorov (1988), Veselov and Veselova (1988), Kozlov (1988), Fedorov and Kozlov (1995), Zenkov and Bloch (2003), Fasso et al. (2019), Jovanović (2015), Fedorov et al. (2015)]. We will consider smooth measures of the form μ = ν n , where n [see (4.5)] is the standard measure on the cotangent bundle T * S and ν is a nonvanishing smooth function, called the density of the measure μ. ...
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... Any further distribution of this work must maintain attribution to the author(s) and the title of the work, journal citation and DOI. moving energy, which is a modification of the generalized energy, is conserved under suitable conditions [8,10,14]. ...
... The steepness assumption is verified using the results of [Nie06,BF17,BFS18]. We remark that at present we are unable to deal with the 3 dimensional case, since the topology of the foliation of the classical integrable Hamiltonian system is very different in the 2 and in the 3 dimensional cases [BF16] and the results by Colin de Verdière [CdV80] do not apply to the 3-d case. ...
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... This is related to the quantum coarsening of the sub-Planck structures and we anticipate that reducing the effective would reveal these finer structures. Such a correspondence has also been observed [101], using a different approach, in a recent study wherein the quantum manifestation of Nekhoroshev stability has been established. Thus, our results in Figure 3(b) show that the generalization may hold for the higher dimensional systems as well. ...
![[object Object]](https://i1.rgstatic.net/ii/profile.image/281399153184779-1444102194289_Q64/D-Sadovskii.jpg)
... Nonintegrable (also known as nonholonomic) constraints are important for engineering applications. One of the application areas of the geometric HJ theory is nonholonomic mechanics [99] and it has been studied by various authors in different perspectives [10,38,75,109]. ...
... These works concern with the specific Kepler Hamiltonion with a small perturbation, which represents the external fields. Recently, Fasso et al. [6] used Nekhoroshey theory [1] to discuss the perturbed hydrogen atom. Our results can be applied in this area. ...
