# Heisenberg model in pseudo-Euclidean spaces

@article{Jovanovic2014HeisenbergMI, title={Heisenberg model in pseudo-Euclidean spaces}, author={B. Jovanovic}, journal={Regular and Chaotic Dynamics}, year={2014}, volume={19}, pages={245-250} }

We construct analogues of the classical Heisenberg spin chain model (or the discrete Neumann system), on pseudo-spheres and light-like cones in the pseudo-Euclidean spaces and show their complete Hamiltonian integrability. Further, we prove that the Heisenberg model on a light-like cone leads to a new example of the integrable discrete contact system.

#### 7 Citations

Heisenberg Model in Pseudo-Euclidean Spaces II

- Mathematics, Physics
- 2014

In the review we describe a relation between the Heisenberg spin chain model on pseudospheres and light-like cones in pseudo-Euclidean spaces and virtual billiards. A geometrical interpretation of… Expand

Contact flows and integrable systems

- Mathematics, Physics
- 2015

Abstract We consider Hamiltonian systems restricted to the hypersurfaces of contact type and obtain a partial version of the Arnold–Liouville theorem: the system need not be integrable on the whole… Expand

Virtual billiards and Heisenberg model in pseudo-Euclidean spaces: discrete Hamiltonian and contact integrability

- Mathematics
- 2015

We consider virtual billiard dynamics within quadrics in pseudo--Euclidean spaces, where in contrast to the usual billiards, the incoming velocity and the velocity after the billiard reflection can… Expand

Virtual billiards in pseudo-Euclidean spaces: discrete Hamiltonian and contact integrability

- Physics, Mathematics
- 2015

The aim of the paper is to unify the efforts in the study of integrable billiards within quadrics in flat and curved spaces and to explore further the interplay of symplectic and contact… Expand

Geodesic and billiard flows on quadrics in pseudo-Euclidean spaces: L-A pairs and Chasles theorem

- Mathematics, Physics
- 2014

In this article we construct L--A representations of geodesic flows on quadrics and of billiard problems within ellipsoids in the pseudo--Euclidean spaces. A geometric interpretation of the… Expand

Billiards on constant curvature spaces and generating functions for systems with constraints

- Physics
- 2017

In this note we consider a method of generating functions for systems with constraints and, as an example, we prove that the billiard mappings for billiards on the Euclidean space, sphere, and the… Expand

Continuous and discrete Neumann systems on Stiefel varieties as matrix generalizations of the Jacobi–Mumford systems

- Physics, Mathematics
- 2015

We study geometric and algebraic geometric properties of the continuous and discrete Neumann systems on cotangent bundles of Stiefel varieties $V_{n,r}$. The systems are integrable in the… Expand

#### References

SHOWING 1-10 OF 25 REFERENCES

Contact flows and integrable systems

- Mathematics, Physics
- 2015

Abstract We consider Hamiltonian systems restricted to the hypersurfaces of contact type and obtain a partial version of the Arnold–Liouville theorem: the system need not be integrable on the whole… Expand

Discrete versions of some classical integrable systems and factorization of matrix polynomials

- Mathematics
- 1991

Discrete versions of several classical integrable systems are investigated, such as a discrete analogue of the higher dimensional force-free spinning top (Euler-Arnold equations), the Heisenberg… Expand

Pseudo-Riemannian geodesics and billiards

- Mathematics
- 2006

Abstract In pseudo-Riemannian geometry the spaces of space-like and time-like geodesics on a pseudo-Riemannian manifold have natural symplectic structures (just like in the Riemannian case), while… Expand

Geometry of Quadrics and Spectral Theory

- Mathematics
- 1980

In this paper we are concerned with integrable Hamiltonian systems. This concept goes back to classical analytical dynamics of the last century. Briefly these are nonlinear systems of ordinary… Expand

Contact complete integrability

- 2010

Complete integrability in a symplectic setting means the existence of a Lagrangian foliation leaf-wise preserved by the dynamics. In the paper we describe complete integrability in a contact set-up… Expand

The Jacobi-Rosochatius Problem on an Ellipsoid: the Lax Representations and Billiards

- Mathematics, Physics
- 2013

In this article, we construct the Lax representations of the geodesic flow, the Jacobi-Rosochatius problem and its perturbations by means of separable polynomial potentials on an ellipsoid. We prove… Expand

Confocal surfaces and integrable billiards on the sphere and in the Lobachevsky space

- Mathematics
- 1990

Abstract Billiard problems for the domains on the sphere and hyperbolic space bounded by the corresponding conical sections are investigated. It is shown that these discrete systems are integrable… Expand

Noncommutative integrability and action-angle variables in contact geometry

- Mathematics, Physics
- 2011

We introduce a notion of the noncommutative integrability within a framework of contact geometry.

Geodesic flows and Neumann systems on Stiefel varieties: geometry and integrability

- Mathematics, Physics
- 2012

We study integrable geodesic flows on Stiefel varieties Vn,r = SO(n)/SO(n−r) given by the Euclidean, normal (standard), Manakov-type, and Einstein metrics. We also consider natural generalizations of… Expand

Ellipsoidal billiards in pseudo-Euclidean spaces and relativistic quadrics

- Mathematics, Physics
- 2012

Abstract We study the geometry of confocal quadrics in pseudo-Euclidean spaces of arbitrary dimension d and any signature, and related billiard dynamics. The goal is to give a complete description of… Expand