scholarly journals Bending Flows for Sums of Rank One Matrices

2005 ◽  
Vol 57 (1) ◽  
pp. 114-158 ◽  
Author(s):  
Hermann Flaschka ◽  
John Millson
Keyword(s):  
Hamiltonian System ◽  
Representation Theory ◽  
Unitary Group ◽  
Dense Subset ◽  
Integrable Hamiltonian System ◽  
Open Dense Subset ◽  
Semiclassical Quantization ◽  
Rank One ◽  
Symplectic Quotients ◽  
Complex Projective

AbstractWe study certain symplectic quotients of n-fold products of complex projective m-space by the unitary group acting diagonally. After studying nonemptiness and smoothness of these quotients we construct the action-angle variables, defined on an open dense subset, of an integrable Hamiltonian system. The semiclassical quantization of this system reporduces formulas from the representation theory of the unitary group.

scholarly journals Genericity of the multibump dynamics for almost periodic Duffing-like systems

1999 ◽  
Vol 129 (5) ◽  
pp. 885-901 ◽  
Author(s):  
Francesca Alessio ◽  
Paolo Caldiroli ◽  
Piero Montecchiari
Keyword(s):  
Variational Methods ◽  
Dense Subset ◽  
Continuous Functions ◽  
Open Dense Subset ◽  
Almost Periodic

In this paper we consider ‘slowly’ oscillating perturbations of almost periodic Duffing-like systems, i.e. systems of the form ü = u − (a(t) + α(wt))W′(u), t ∈ ℝ, u ∈ ℝN, where W ∈ C2N(ℝN, ℝ) is superquadratic and a and α are positive and almost periodic. By variational methods, we prove that if w > 0 is small enough, then the system admits a multibump dynamics. As a consequence we get that the system ü = u − a(t)W′(u), t ∈ ℝ, u ∈ ℝN, admits multibump solutions whenever a belongs to an open dense subset of the set of positive almost periodic continuous functions.

Multisymplectic Reduction for Proper Actions

2004 ◽  
Vol 56 (3) ◽  
pp. 638-654 ◽  
Author(s):  
Jędrzej Śniatycki
Keyword(s):  
Symmetry Group ◽  
Dense Subset ◽  
Variational Problems ◽  
Open Dense Subset ◽  
Proper Action ◽  
Partial Derivatives ◽  
Proper Actions ◽  
Reduced Equations ◽  
Definition Of

AbstractWe consider symmetries of the Dedonder equation arising from variational problems with partial derivatives. Assuming a proper action of the symmetry group, we identify a set of reduced equations on an open dense subset of the domain of definition of the fields under consideration. By continuity, the Dedonder equation is satisfied whenever the reduced equations are satisfied.

scholarly journals QUANTIZATION OF A MODULI SPACE OF PARABOLIC HIGGS BUNDLES

2004 ◽  
Vol 15 (09) ◽  
pp. 907-917 ◽  
Author(s):  
INDRANIL BISWAS ◽  
AVIJIT MUKHERJEE
Keyword(s):  
Riemann Surface ◽  
Moduli Space ◽  
Symplectic Form ◽  
Dense Subset ◽  
Projective Structure ◽  
Open Dense Subset ◽  
Higgs Bundles ◽  
Smooth Variety ◽  
Structure Formula

Let [Formula: see text] be a moduli space of stable parabolic Higgs bundles of rank two over a Riemann surface X. It is a smooth variety defined over [Formula: see text] equipped with a holomorphic symplectic form. Fix a projective structure [Formula: see text] on X. Using [Formula: see text], we construct a quantization of a certain Zariski open dense subset of the symplectic variety [Formula: see text].

On the Diffusion Along Resonant Lines in Continuous and Discrete Dynamical Systems

2003 ◽  
Vol 17 (22n24) ◽  
pp. 3964-3976
Author(s):  
Claude Froeschlé ◽  
Elena Lega
Keyword(s):  
Dynamical Systems ◽  
Hamiltonian System ◽  
Coupling Parameter ◽  
A Priori ◽  
Integrable Hamiltonian System ◽  
Discrete Dynamical Systems ◽  
Nekhoroshev Theorem ◽  
Symplectic Map ◽  
Quasi Integrable Hamiltonian System

We detect and measure diffusion along resonances in a discrete symplectic map for different values of the coupling parameter. Qualitatively and quantitatively the results are very similar to those obtained for a quasi-integrable Hamiltonian system, i.e. in agreement with Nekhoroshev predictions, although the discrete mapping does not fulfill completely, a priori, the conditions of the Nekhoroshev theorem.

scholarly journals Orbital Structure of the Two Fixed Centres Problem

1999 ◽  
Vol 172 ◽  
pp. 463-464
Author(s):  
A. Cordero ◽  
J. Martínez Alfaro ◽  
P. Vindel
Keyword(s):  
Hamiltonian System ◽  
Periodic Orbits ◽  
Degrees Of Freedom ◽  
Artificial Satellite ◽  
Equilibrium Points ◽  
Integrable Hamiltonian System ◽  
Three Dimensional ◽  
First Integral ◽  
Material Point ◽  
Dimensional Sphere

The set of orbits of the Two Fixed Centres problem has been known for a long time (Chartier, 1902, 1907; Pars, 1965), since it is an integrable Hamiltonian system.We consider a plane that contains the fixed masses. Denote by φ the angle denned by this plane and the one that contains also the third body. The momentum pφ is a first integral of the system and when pφ is different from zero, the manifold generated by the generalized coordinates and momenta are two copies of the three-dimensional sphere S3. If pφ = 0, that is to say when the planet crosses the line joining both suns, the motion is restricted to a planar one. All the equilibrium points appears in this case and therefore the phase spaces are more complex. We restrict our attention to this case which has two degrees of freedom.It is again a Bott-integrable Hamiltonian system. The set of periodic orbits of this systems can be studied from a subset of them, the Non-Singular Morse-Smale type orbits (see Casasayas, 1992). It is proved in Campos (1997) that a small perturbation of a Bott-integrable Hamiltonian system transforms it into a Non-Singular Morse-Smale system. The NMS periodic orbits belong to both the NMS system and the Hamiltonian one. Moreover, The NMS p.o. can be continued to nearly Hamiltonian systems. For instance, in our case to the Restricted Three Body Problem and in the study of the motion of a material point moving inside the gravitational field generated by two stars. This approximation is also useful when the motion of an artificial satellite around a spheroidal body is considered.

The Dynamical Mordell–Lang Conjecture for Skew-Linear Self-Maps. Appendix by Michael Wibmer

2018 ◽  
Vol 2020 (21) ◽  
pp. 7433-7453
Author(s):  
Dragos Ghioca ◽  
Junyi Xie
Keyword(s):  
Linear Transformation ◽  
Difference Equations ◽  
Dense Subset ◽  
Closed Field ◽  
Open Dense Subset ◽  
Irreducible Curve ◽  
Algebraically Closed Field ◽  
Set Of Points ◽  
Dense Set

Abstract Let $k$ be an algebraically closed field of characteristic $0$, let $N\in{\mathbb{N}}$, let $g:{\mathbb{P}}^1{\longrightarrow } {\mathbb{P}}^1$ be a nonconstant morphism, and let $A:{\mathbb{A}}^N{\longrightarrow } {\mathbb{A}}^N$ be a linear transformation defined over $k({\mathbb{P}}^1)$, that is, for a Zariski-open dense subset $U\subset{\mathbb{P}}^1$, we have that for $x\in U(k)$, the specialization $A(x)$ is an $N$-by-$N$ matrix with entries in $k$. We let $f:{\mathbb{P}}^1\times{\mathbb{A}}^N{\dashrightarrow } {\mathbb{P}}^1\times{\mathbb{A}}^N$ be the rational endomorphism given by $(x,y)\mapsto (\,g(x), A(x)y)$. We prove that if $g$ induces an automorphism of ${\mathbb{A}}^1\subset{\mathbb{P}}^1$, then each irreducible curve $C\subset{\mathbb{A}}^1\times{\mathbb{A}}^N$ that intersects some orbit $\mathcal{O}_f(z)$ in infinitely many points must be periodic under the action of $f$. Furthermore, in the case $g:{\mathbb{P}}^1{\longrightarrow } {\mathbb{P}}^1$ is an endomorphism of degree greater than $1$, then we prove that each irreducible subvariety $Y\subset{\mathbb{P}}^1\times{\mathbb{A}}^N$ intersecting an orbit $\mathcal{O}_f(z)$ in a Zariski dense set of points must be periodic. Our results provide the desired conclusion in the Dynamical Mordell–Lang Conjecture in a couple new instances. Moreover, our results have interesting consequences toward a conjecture of Rubel and toward a generalized Skolem–Mahler–Lech problem proposed by Wibmer in the context of difference equations. In the appendix it is shown that the results can also be used to construct Picard–Vessiot extensions in the ring of sequences.

scholarly journals BINARY NONLINEARIZATION OF THE SUPER AKNS SYSTEM

2008 ◽  
Vol 22 (04) ◽  
pp. 275-288 ◽  
Author(s):  
JINGSONG HE ◽  
JING YU ◽  
YI CHENG ◽  
RUGUANG ZHOU
Keyword(s):  
Hamiltonian System ◽  
Spectral Problem ◽  
Integrable Hamiltonian System ◽  
Integrals Of Motion ◽  
Finite Dimensional ◽  
Akns System ◽  
Hamiltonian Forms

We establish the binary nonlinearization approach of the spectral problem of the super AKNS system, and then use it to obtain the super finite-dimensional integrable Hamiltonian system in the supersymmetry manifold ℝ4N|2N. The super Hamiltonian forms and integrals of motion are given explicitly.

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