Table 1 - uploaded by James Montaldi
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Representatives of conjugacy classes of proper isotropy subgroups of the action of D, X S' on C2 given by (4.1).
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The authors analyse the existence of nonlinear normal modes of a (nonlinear) Hamiltonian system, i.e. periodic solutions that approximate periodic solutions of the system linearised around an elliptic (and semisimple) equilibrium point. In particular they consider systems with symmetry, including time-reversible symmetry which involves an antisympl...
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Citations
... We define the ω-semisymplectic group n = Sp ω (n; R)∪ Sp −1 ω (n; R), where Sp −1 ω (n; R) is the set of antisymplectic matrices, which provides all algebraic information necessary for the study of symplectic and antisymplectic linear operators. These maps arise naturally in physical systems and are closely related to semisymplectic actions, which appear in several studies involving symmetric Hamiltonian systems, as for instance in [1][2][3][4]11,12,18]. In [2], we characterize subgroups of the ω-semisymplectic group in terms of a semisymplectic action. ...
The purpose of this paper is to present an algebraic theoretical basis for the study of ω\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\omega $$\end{document}-Hamiltonian vector fields defined on a symplectic vector space (V,ω)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(V,\omega )$$\end{document} with respect to coordinates that are not necessarily symplectic. We introduce the concepts of ω\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\omega $$\end{document}-symplectic and ω\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\omega $$\end{document}-semisymplectic groups, and describe some of their properties that may not coincide with the classical context. We show that the Lie algebra of such groups is a useful tool in the recognition and construction of ω\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\omega $$\end{document}-Hamiltonian vector fields.
... Particularly interesting is to understand under what conditions the frequencies have to fulfill in order for a system to have a number of periodic orbits exceeding the esitimation given in [64]. There have been further generalizations to [64] and the reader could refer to [45,46]. ...
... This is a tool used in [44] to study the Floquet operator in a neighbourhood of the RPO. For applications in the study of nonlinear normal modes and stability see [45,46]. In [67] it is shown that such construction can be used to decompose the Poincaré section in a part which is tangent to the conserved momentum, another part tangent to shape and a part parameterizing the momentum. ...
This is a draft of new edition Chapter (2008 first edition) written for the Springer Encyclopedia of Complexity Science. The Chapter contains a general description of the study of periodic motions in Hamiltonian systems. Different methods and techniques are described through examples.
In particular the Chapter collected some proofs and calculation methods which are scattered in the vast literature on the subject.
... When one finally has a G × S 1 invariant polynomial system the parameter dependent equation for the stationary points of the S 1 -reduced system then gives the bifurcation equation. For S 1 -symmetric systems these ideas were introduced in [15], [65], and for G × S 1 invariant systems in [39,40,41]. Also see [21,22]. ...
In this note we will consider reduction techniques for Hamiltonian systems that are invariant under the action of a compact Lie group $G$ acting by symplectic diffeomorphisms and the related work on stability of relative equilibria. We will focus on reduction by invariants in which case it is possible to describe a reduced phase space within the orbit space by constructing an orbit map using a Hilbert basis of invariants for the symmetry group $G$. Results considering the stratification, foliation and fibration of the phase space and the orbit space are considered. Finally some remarks are made concerning relative equilibria and bifurcations of periodic solutions. We will combine results from a wide variety of papers.
... where the bar denotes the closure of the set and L is the linearization of the vector field at the origin. In this case, the normal form process introduces additional symmetries into the problem since every vector field is formally conjugated to a vector field with symmetry group S. Several works deal with vector fields that present symmetric geometric configurations, as we can see in [4,5,9,15,18]. In particular, in the same way as Elphick et al. [14] and based on algebraic invariant theory, Baptistelli, Manoel and Zeli present a method ( [4,Theorem 4.7]) to obtain normal forms of a reversible equivariant vector field that preserve the symmetries of the original vector field. ...
... Given (V, ω) a symplectic real vector space of dimension 2n and H : V → R a smooth function, the ω-Hamiltonian vector field associated with H can be written as X(x) = ([ω] −1 ) T ∇H(x) for all x ∈ V , where [ω] denotes the matrix of ω relative to a basis of V (see Definition 2.1). In symmetric Hamiltonian context, several works have contributed in different aspects, such as [1,10,11,18]. In [18], Montaldi, Roberts and Stewart deal with Hamiltonian systems on (R 2n , ω 0 ), where ω 0 is the canonical bilinear form, under an orthogonal action of a Lie group that admits an involutory reversible symmetry. ...
... In symmetric Hamiltonian context, several works have contributed in different aspects, such as [1,10,11,18]. In [18], Montaldi, Roberts and Stewart deal with Hamiltonian systems on (R 2n , ω 0 ), where ω 0 is the canonical bilinear form, under an orthogonal action of a Lie group that admits an involutory reversible symmetry. In [11], Buzzi and Teixeira also analyze the dynamics of reversible Hamiltonian vector fields on (R 2n , ω 0 ) with respect to a symplectic involution. ...
In this paper, we present algebraic tools to obtain normal forms of $\omega$-Hamiltonian vector fields under a semisymplectic action of a Lie group, by taking into account the symmetries and reversing symmetries of the vector field. The normal forms resulting from the process preserve the Hamiltonian condition and the types of symmetries of the original vector field. Our techniques combine the classical method of normal forms of Hamiltonian vector fields with the invariant theory of groups.
... When the oscillator is perturbed, the orbits will have different evolutions: let us still call normal modes the solutions, if they exist, corresponding to J 1 = 0 and J 2 = 0 (these are nonlinear normal modes in the sense of Moser & Weinstein [53,79]; see also [51,52]), and orbits in generic position those for which both J 1 and J 2 are nonzero. ...
From the point of view of perturbation theory, (perturbations of) near-resonant systems are plagued by small denominators. These do not affect (perturbations of) fully resonant systems; so it is in many ways convenient to approximate near resonant systems as fully resonant ones, which correspond to considering the "detuning" as a perturbation itself. This approach has proven very fruitful in Classical Mechanics, but it is also standard in (perturbations of) Quantum Mechanical systems. Actually, its origin may be traced back (at least) to the Rayleigh-Ritz method for computing eigenvalues and eigenvectors of perturbed matrix problems. We will discuss relations between these approaches, and consider some case study models in the different contexts.
... In the case of non-linear Hamiltonian systems, there may be certain families of periodic oscillations that tend to the normal linear modes of the respective dynamical system linearised around the equilibrium position (these are what Rosenberg would later call 'similar' modes). These families of motions are the NNMs [82]. ...
This paper addresses a detailed literature review on non-linear modes of vibration and its role in obtaining more efficient reduced-order models, as compared to those obtained by Galerkin's method using classical linear modes. Examples that emphasise the superior quality of the former models are supplied. Regarding the review on non-linear modes, a chronological development is presented, stressing the main stages of definitions, properties, methods for evaluation and application of non-linear modes to engineering problems, from Poincaré to Liapunov, from Rosenberg and Vakakis to Shaw and Pierre, from past to present.
... Thus, the ω-semisymplectic group Ω n = Sp ω (n; R)∪ Sp −1 ω (n; R) provides all algebraic information that is necessary for our study of linear symplectic and antisymplectic maps. These types of maps arise naturally in physical systems and are closely related to semisymplectic actions, which appear in several studies involving symmetric Hamiltonian systems, as for instance in [1,2,3,4,13,14,18]. ...
... When X is a Hamiltonian vector field defined on the standard symplectic space (R 2n , ω 0 ), it is possible to prove that S is a subgroup of Sp(n; R). In particular, the authors in [14] use this property to obtain some results for Hamiltonian vector fields. In a more general context, we characterize when S is a subgroup of Sp ω (n; R) in terms of the matrices in sp ω (n; R). ...
... , k. So (14) (dX j x ) T [ω] + [ω]dX j x = 0, for each j = 1, . . . , k. ...
The purpose of this paper is presenting a theoretical basis for the study of $\omega$-Hamiltonian vector fields in a more general approach than the classical one. We introduce the concepts of $\omega$-symplectic group and $\omega$-semisymplectic group, and describe some of their properties. We show that the Lie algebra of such groups is a useful tool in the recognition of an $\omega$-Hamiltonian vector field defined on a symplectic vector space $(V,\omega)$ with respect to coordinates that are not necessarily symplectic.
... The first works of Montaldi on G-Hamiltonian systems are a series of articles with long time collaborators M. Roberts and I. Stewart during the late 80's [21], [22] and [23]. They are devoted to the study of a closely related subject: the equivariant version of the well known Moser-Weinstein theorem [29] and [40]. ...
... The concept of normal modes is a fundamental one in the study of Hamiltonian dynamical systems [1,11,13,15]; it is based on the quadratic part of the Taylor expansion of the Hamiltonian around a non-degenerate (isolated and hyperbolic) stable equilibrium, and under certain fairly general assumptions it can be conveniently employed also in considering higher order expansions of the Hamiltonian. That is, under such assumptions (including a non-resonance condition) normal modes persist, at least locally, when one considers also higher order terms -or for nonlinear Hamiltonian dynamics [17,29] -as also studied by James in a series of papers [18,19]. ...
... 2. The term nonlinear normal mode (NNM) was introduced by Montaldi et al. [57,58]. Further discussion and examples can be found in [32,36,59,60]. ...
We analyse qualitatively the bending vibrational polyads of the acetylene molecule (C H ) in the approximation of the resonant oscillator with axial symmetry using an effective vibrational Hamiltonian which reproduces bending vibrational energy levels computed by Michel Herman and coworkers [M. Herman, A. Campargue, M.I.E. Idrissi, and J.V. Auwera, J. Phys. Chem. Ref. Data 32 (3), 921 (2003). doi: 10.1063/1.1531651]. We explain how the classical limit of this quantum system for the total vibrational angular momentum is equivalent to a reduced perturbed Keplerian system on the classical phase space , such as the hydrogen atom in external electric and magnetic fields in the Kustaanheimo–Stiefel formalism. In particular, bending vibrational -polyads of C H correspond to the n-shells of the perturbed hydrogen atom. Within this approach, using the techniques developed for the Keplerian systems and methods of the qualitative theory, we account concisely for all series of bifurcations of the classical nonlinear normal modes and their manifestations in the quantum energy level spectrum described by our predecessors [V. Tyng and M.E. Kellman, J. Phys. Chem. B 110 (38), 18859 (2006). doi: 10.1021/jp057357f]. In addition to local oscillator approximations near stable equilibrium points, notably the local modes discussed by Rose and Kellman [J. Chem. Phys. 105 (24), 10743 (1996). doi: 10.1063/1.472882] and Jacobson et al. [J. Chem. Phys. 109 (1), 121 (1998). doi: 10.1063/1.476529; J. Chem. Phys. 111, 600 (1999). doi: 10.1063/1.479341; J. Chem. Phys. 110, 845 (1999). doi: 10.1063/1.478052], we introduce two new global integrable approximations, and confirm them by constructing respective two regular complementary lattices of quantum states within one -polyad. From the stratification of the phase space, we uncover the geometrical meaning of the corresponding new good quantum numbers and define new kind of wavefunction localisation in the neighbourhood of two spheres in the four-dimensional space. Furthermore, we use our two approximate lattices to uncover quantum monodromy of the system through the evolution of an elementary quantum cell along a closed path in the images of the nearly integrable energy-momentum maps.
