scholarly journals SUPERINTEGRABLE SYSTEMS, MULTI-HAMILTONIAN STRUCTURES AND NAMBU MECHANICS IN AN ARBITRARY DIMENSION

2004 ◽  
Vol 19 (03) ◽  
pp. 393-409 ◽  
Author(s):  
A. TEĞMEN ◽  
A. VERÇIN
Keyword(s):  
Arbitrary Dimension ◽  
Functional Independence ◽  
Poisson Structures ◽  
Algebraic Condition ◽  
Superintegrable System ◽  
Hamiltonian Structures ◽  
Superintegrable Systems ◽  
Constants Of Motion ◽  
Nambu Bracket ◽  
Moser System

A general algebraic condition for the functional independence of 2n-1 constants of motion of an n-dimensional maximal superintegrable Hamiltonian system has been proved for an arbitrary finite n. This makes it possible to construct, in a well-defined generic way, a normalized Nambu bracket which produces the correct Hamiltonian time evolution. Existence and explicit forms of pairwise compatible multi-Hamiltonian structures for any maximal superintegrable system have been established. The Calogero–Moser system, motion of a charged particle in a uniform perpendicular magnetic field and Smorodinsky–Winternitz potentials are considered as illustrative applications and their symmetry algebras as well as their Nambu formulations and alternative Poisson structures are presented.

scholarly journals On the Integrable Deformations of the Maximally Superintegrable Systems

Symmetry ◽  
2021 ◽  
Vol 13 (6) ◽  
pp. 1000
Author(s):  
Cristian Lăzureanu
Keyword(s):  
Differential Equations ◽  
Arbitrary System ◽  
Superintegrable System ◽  
First Order ◽  
Integrable Deformation ◽  
Superintegrable Systems ◽  
Constants Of Motion ◽  
Integrable Deformations ◽  
Autonomous Differential Equations ◽  
New System

In this paper, we present the integrable deformations method for a maximally superintegrable system. We alter the constants of motion, and using these new functions, we construct a new system which is an integrable deformation of the initial system. In this manner, new maximally superintegrable systems are obtained. We also consider the particular case of Hamiltonian mechanical systems. In addition, we use this method to construct some deformations of an arbitrary system of first-order autonomous differential equations.

scholarly journals Carter's constant and superintegrability

2018 ◽  
Vol 27 (07) ◽  
pp. 1850066
Author(s):  
Payel Mukhopadhyay ◽  
K. Rajesh Nayak
Keyword(s):  
Dynamical System ◽  
Degrees Of Freedom ◽  
Three Dimensional ◽  
Two Dimensional ◽  
Integrals Of Motion ◽  
Superintegrable System ◽  
Simple Observation ◽  
Superintegrable Systems ◽  
Axisymmetric Spacetime ◽  
Three Degrees Of Freedom

Carter's constant is a nontrivial conserved quantity of motion of a particle moving in stationary axisymmetric spacetime. In the version of the theorem originally given by Carter, due to the presence of two Killing vectors, the system effectively has two degrees of freedom. We propose an extension to the first version of Carter's theorem to a system having three degrees of freedom to find two functionally independent Carter-like integrals of motion. We further generalize the theorem to a dynamical system with [Formula: see text] degrees of freedom. We further study the implications of Carter's constant to superintegrability and present a different approach to probe a superintegrable system. Our formalism gives another viewpoint to a superintegrable system using the simple observation of separable Hamiltonian according to Carter's criteria. We then give some examples by constructing some two-dimensional superintegrable systems based on this idea and also show that all three-dimensional simple classical superintegrable potentials are also Carter separable.

Dirac equation with anisotropic oscillator, quantum E3′ and Holt superintegrable potentials and relativistic generalized Yang–Coulomb monopole system

2016 ◽  
Vol 14 (01) ◽  
pp. 1750004 ◽  
Author(s):  
Vahid Mohammadi ◽  
Alireza Chenaghlou
Keyword(s):  
Dirac Equation ◽  
Relativistic Energy ◽  
Integrals Of Motion ◽  
Superintegrable System ◽  
Energy Eigenvalues ◽  
System A ◽  
Superintegrable Systems ◽  
Casimir Operators ◽  
Oscillator Quantum ◽  
Anisotropic Oscillator

The two-dimensional Dirac equation with spin and pseudo-spin symmetries is investigated in the presence of the maximally superintegrable potentials. The integrals of motion and the quadratic algebras of the superintegrable quantum [Formula: see text], anisotropic oscillator and the Holt potentials are studied. The corresponding Casimir operators and the structure functions of the mentioned superintegrable systems are found. Also, we obtain the relativistic energy spectra of the corresponding superintegrable systems. Finally, the relativistic energy eigenvalues of the generalized Yang–Coulomb monopole (YCM) superintegrable system (a [Formula: see text] non-Abelian monopole) are calculated by the energy spectrum of the eight-dimensional oscillator which is dual to the former system by Hurwitz transformation.

scholarly journals Stability and Energy-Casimir Mapping for Integrable Deformations of the Kermack-McKendrick System

2018 ◽  
Vol 2018 ◽  
pp. 1-9 ◽  
Author(s):  
Cristian Lăzureanu ◽  
Camelia Petrişor
Keyword(s):  
Integrable Systems ◽  
Equilibrium Points ◽  
Poisson System ◽  
Poisson Structures ◽  
Integrable Deformation ◽  
Constants Of Motion ◽  
Integrable Deformations ◽  
Dynamical Properties ◽  
The Stability ◽  
Mckendrick Model

Integrable deformations of a Hamilton-Poisson system can be obtained altering its constants of motion. These deformations are integrable systems that can have various dynamical properties. In this paper, we give integrable deformations of the Kermack-McKendrick model for epidemics, and we analyze a particular integrable deformation. More precisely, we point out two Poisson structures that lead to infinitely many Hamilton-Poisson realizations of the considered system. Furthermore, we study the stability of the equilibrium points, we give the image of the energy-Casimir mapping, and we point out some of its properties.

SUPER-DISPERSIVE LONG WAVE HIERARCHY AND ITS SUPER-HAMILTONIAN STRUCTURES

2009 ◽  
Vol 23 (24) ◽  
pp. 2899-2905
Author(s):  
HONG-WEI YANG ◽  
HUAN-HE DONG ◽  
ZHU LI
Keyword(s):  
Hamiltonian Structure ◽  
The Other ◽  
Hamiltonian Structures ◽  
Long Wave ◽  
Superintegrable Systems

Super-dispersive long wave hierarchy is obtained by use of the Lie super-algebra B(0, 1), then the super-Hamiltonian structure of the above system is given by the associated supertrace identity. The method can be used to produce the super-Hamiltonian structures of the other superintegrable systems.

scholarly journals Stäckel Equivalence of Non-Degenerate Superintegrable Systems, and Invariant Quadrics

2021 ◽  
Author(s):  
Andreas Vollmer ◽  
Keyword(s):  
Equivalence Class ◽  
Second Order ◽  
Coupling Constant ◽  
Superintegrable System ◽  
Superintegrable Systems ◽  
Dimension 2

A non-degenerate second-order maximally conformally superintegrable system in dimension 2 naturally gives rise to a quadric with position dependent coefficients. It is shown how the system's Stäckel class can be obtained from this associated quadric.The Stäckel class of a second-order maximally conformally superintegrable system is its equivalence class under Stäckel transformations, i.e., under coupling-constant metamorphosis.

scholarly journals Noncompact CPN as a phase space of superintegrable systems

2021 ◽  
Vol 36 (08n09) ◽  
pp. 2150055
Author(s):  
Erik Khastyan ◽  
Armen Nersessian ◽  
Hovhannes Shmavonyan
Keyword(s):  
Phase Space ◽  
Complex Projective Space ◽  
Coulomb Systems ◽  
Kähler Structure ◽  
Higher Dimensional ◽  
Superintegrable Systems ◽  
Constants Of Motion ◽  
Complex Projective ◽  
Klein Model

We propose the description of superintegrable models with dynamical [Formula: see text] symmetry, and of the generic superintegrable deformations of oscillator and Coulomb systems in terms of higher-dimensional Klein model (the noncompact analog of complex projective space) playing the role of phase space. We present the expressions of the constants of motion of these systems via Killing potentials defining the [Formula: see text] isometries of the Kähler structure.

scholarly journals Duality relation among the Hamiltonian structures of a parametric coupled Korteweg-de Vries system

Open Physics ◽  
2016 ◽  
Vol 14 (1) ◽  
pp. 95-105 ◽  
Author(s):  
Alvaro Restuccia ◽  
Adrián Sotomayor
Keyword(s):  
Hamiltonian Structure ◽  
Coupled System ◽  
Weak Limit ◽  
Constrained Systems ◽  
Poisson Structures ◽  
Strong Limit ◽  
Field Equations ◽  
Algebraic Construction ◽  
Hamiltonian Structures ◽  
Master System

AbstractWe obtain the full Hamiltonian structure for a parametric coupled KdV system. The coupled system arises from four different real basic lagrangians. The associated Hamiltonian functionals and the corresponding Poisson structures follow from the geometry of a constrained phase space by using the Dirac approach for constrained systems. The overall algebraic structure for the system is given in terms of two pencils of Poisson structures with associated Hamiltonians depending on the parameter of the Poisson pencils. The algebraic construction we present admits the most general space of observables related to the coupled system. We then construct two master lagrangians for the coupled system whose field equations are the ε-parametric Gardner equations obtained from the coupled KdV system through a Gardner transformation. In the weak limit ε → 0 the lagrangians reduce to the ones of the coupled KdV system while, after a suitable redefinition of the fields, in the strong limit ε → ∞ we obtain the lagrangians of the coupled modified KdV system. The Hamiltonian structures of the coupled KdV system follow from the Hamiltonian structures of the master system by taking the two limits ε → 0 and ε → ∞.

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