1,721,024 research outputs found
Quasi-periodicity of motions and complete integrability of Hamiltonian systems
No full textAbstract: Consider a Hamiltonian system with d degrees of freedom whose motions are all linear on tori of some fixed dimension n less than or equal to d; 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
LIE SERIES METHOD FOR VECTOR-FIELDS AND HAMILTONIAN PERTURBATION-THEORY
No full textAbstract: 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 informations on the normal form. Moreover, it leads to estimates which are better and/or simpler than those of the scalar Lie methods. The perturbation method is presented with reference to two model problems, both pertaining to the realm of the well known Nekhoroshev theorem: the confining of actions for exponentially long times in a system of coupled harmonic oscillators, and an application to the so called problem of the realization of a holonomic constraint in classical mechanics
Comparison of splitting algorithms for the rigid body
No full textAbstract: 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 algorithms can produce energy errors which differ by several orders of magnitude. As a byproduct of this analysis we remark that, for the special case of a flat rigid body with moments of inertia proportional to (1, 0.75 0.25), one of the considered algorithms is in fact of order four
The Euler-Poinsot top: A non-commutatively integrable system without global action-angle coordinates
No full textAbstract: 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 of the geometry of integrable degenerate Hamiltonian systems. In this way, we get a global geometric picture of the Euler-Poinsot system which fully accounts for its degeneracy through the (Poisson) structure? of the base of the fibration by the two-dimensional invariant tori. In particular, we explain in this way why this system does not possess global 'generalized' action-angle coordinates: the obstructions are the topological non-triviality of the fibration by the invariant two-dimensional tori and the compactness of the symplectic leaves of its base manifold. We also compare this description with the usual description based on the notion of complete integrability, and we remark that, as a general fact, such a common approach fails to provide a natural, thorough description of degenerate systems
HAMILTONIAN PERTURBATION-THEORY ON A MANIFOLD
No full textAbstract: 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 normal forms within the domain of a local coordinate system, are not immediately usable to study perturbations of these systems, since degeneracy makes impossible to control that the system does not fall into a singularity of the coordinates. To overcome this difficulty, we develop a 'global' formulation of Hamiltonian perturbation theory, in which the normal forms are globally defined on the phase space manifold. The key for this study lies in the geometry of the fibration by the invariant tori of an integrable degenerate Hamiltonian system, which is described by some generalizations of the Liouville-Arnol'd theorem and is reviewed in the paper. As an application, we provide a 'global' formulation of Nekhoroshev's theorem on the stability for exponentially long times
On a relation among Lie series
No full textAbstract: 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 resembles the algorithm of Giorgilli and Galgani, and differs from it only in an ordering property. The relation with the algorithms of Kamel and Henrard for Deprit's direct and inverse series is also discussed
Perturbations of Superintegrable Hamiltonian Systems
Get PDFThis is a review of the structure of superintegrable (or noncommutatively integrable) Hamiltonian systems and of the dynamics of their perturbations
Superintegrable Hamiltonian systems: Geometry and perturbations
Get PDFMany 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 n < d. A thorough comprehension of these systems requires a description which goes beyond the standard notion of Liouville - Arnold integrability, that is, the existence of an invariant fibration by Lagrangian tori. Instead, the natural object to look at is formed by both the fibration by the ( isotropic) invariant tori and by its (coisotropic) polar foliation, which together form what in symplectic geometry is called a 'dual pair', or 'bifoliation', or 'bifibration'. We review this geometric structure, relating it to the dynamical properties of superintegrable systems and pointing out its importance for a thorough understanding of these systems
Classical Freezing of Plane Rotations - A Proof of the Boltzmann-jeans Conjecture
No full textUsing 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 velocity omega is initially large, then the energy exchanges are small, O(omega-1), for times growing exponentially with omega, \t\ approximately exp omega. We also deduce that in a scattering process from a (smooth) potential barrier, the overall change in the rotational energy of the incoming body is exponentially small in omega, E approximately exp(- omega). The results are interpreted in the light of an old conjecture by Boltzmann and Jeans on the existence of very large time scales for equilibrium in statistical systems containing high-frequency degrees of freedom (purely classical "freezing" of the high-frequency degrees of freedom); the rotating object is, in this interpretation, a (classical) molecule, which moves in an external field, or collides with the wall of a container. Two different limits of large omega are considered, namely the limit of large rotational energy, and (as is interesting for the molecular interpretation) the limit of point mass, at finite rotational energy
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