scholarly journals On the Extended-Hamiltonian Structure of Certain Superintegrable Systems on Constant-Curvature Riemannian and Pseudo-Riemannian Surfaces

2020 ◽  
Author(s):  
Claudia Maria Chanu ◽  
◽  
Giovanni Rastelli ◽  
◽  
◽  
...  
Keyword(s):  
Constant Curvature ◽  
Hamiltonian Structure ◽  
Riemannian Surfaces ◽  
Superintegrable Systems

MULTIPLE LOCAL LAGRANGIANS FOR n-COMPONENT SUPER-KORTEWEG–DE VRIES-TYPE BI-HAMILTONIAN SYSTEMS

2010 ◽  
Vol 25 (05) ◽  
pp. 1069-1078 ◽  
Author(s):  
ÖMER OĞUZ ◽  
DEVRIM YAZICI
Keyword(s):  
Hamiltonian Systems ◽  
Hamiltonian Structure ◽  
Velocity Fields ◽  
Kinetic Term ◽  
Lagrangian Formalism ◽  
Momentum Maps ◽  
Miura Transformation ◽  
De Vries ◽  
Superintegrable Systems

The multiple Lagrangian formalism is constructed for n-component Korteweg–de Vries (KdV) type superintegrable systems. They all admit bi-Hamiltonian structure. The first two Lagrangians are local and degenerate. They contain Clebsch potentials for velocity fields and momentum maps in kinetic term. The first local Lagrangian for n-component supermodified KdV (smKdV) is also obtained by employing the multicomponent super-Miura transformation.

scholarly journals Quadratic algebra contractions and second-order superintegrable systems

2014 ◽  
Vol 12 (05) ◽  
pp. 583-612 ◽  
Author(s):  
Ernest G. Kalnins ◽  
W. Miller
Keyword(s):  
Constant Curvature ◽  
Two Dimensions ◽  
Second Order ◽  
Quadratic Algebra ◽  
Quadratic Algebras ◽  
Physical Systems ◽  
Special Cases ◽  
Superintegrable Systems ◽  
Wilson Polynomials ◽  
Symmetry Algebras

Quadratic algebras are generalizations of Lie algebras; they include the symmetry algebras of second-order superintegrable systems in two dimensions as special cases. The superintegrable systems are exactly solvable physical systems in classical and quantum mechanics. For constant curvature spaces, we show that the free quadratic algebras generated by the first- and second-order elements in the enveloping algebras of their Euclidean and orthogonal symmetry algebras correspond one-to-one with the possible superintegrable systems with potential defined on these spaces. We describe a contraction theory for quadratic algebras and show that for constant curvature superintegrable systems, ordinary Lie algebra contractions induce contractions of the quadratic algebras of the superintegrable systems that correspond to geometrical pointwise limits of the physical systems. One consequence is that by contracting function space realizations of representations of the generic superintegrable quantum system on the 2-sphere (which give the structure equations for Racah/Wilson polynomials) to the other superintegrable systems one obtains the full Askey scheme of orthogonal hypergeometric polynomials.

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 Superintegrable systems on spaces of constant curvature

2014 ◽  
Vol 346 ◽  
pp. 91-102 ◽  
Author(s):  
Cezary Gonera ◽  
Magdalena Kaszubska
Keyword(s):  
Constant Curvature ◽  
Spaces Of Constant Curvature ◽  
Superintegrable Systems

Center of Mass Theorem and the Separation of Variables for Two-Body System on a Three-Dimensional Sphere

2020 ◽  
Vol 23 (3) ◽  
pp. 306-311
Author(s):  
Yu. Kurochkin ◽  
Dz. Shoukavy ◽  
I. Boyarina
Keyword(s):  
Constant Curvature ◽  
Jacobi Equation ◽  
Separation Of Variables ◽  
Center Of Mass ◽  
Three Dimensional ◽  
Relative Distance ◽  
Dimensional Sphere ◽  
Body System ◽  
Spaces Of Constant Curvature ◽  
Hamilton Jacobi Equation

The immobility of the center of mass in spaces of constant curvature is postulated based on its definition obtained in [1]. The system of two particles which interact through a potential depending only on the distance between particles on a three-dimensional sphere is considered. The Hamilton-Jacobi equation is formulated and its solutions and trajectory equations are found. It was established that the reduced mass of the system depends on the relative distance.

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