Heisenberg Model in Pseudo-Euclidean Spaces II

Abstract

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 the integrals associated to a family of confocal quadrics is given, analogous to Moser's geometrical interpretation of the integrals of the Neumann system on the sphere.2010 Mathematics Subject Classification. 70H06, 37J35, 37J55, 70H45. Key words and phrases. discrete systems with constraints, contact integrability, billiards, Neumann and Heisenberg systems.1 A draft of Section 2 of the current paper is given as the Section 5 in the first arXive version of(see [14]). 3 It is a completely integrable discrete Hamiltonian system. For J 2 j = J 2 i , the integrals can be written in the form. . , n, with the relation i f i ≡ c 2 among them. Furthermore, on the light-like cone, the mapping Φ leads to an integrable contact system as well (see [14]).2 We hope that it will be clear from the context when k denotes the discrete time, and when the signature of the metric. 3 Actually, the function q k , J −1 q k+1 is the first integral [14], and so the condition q k , J −1 q k+1 = 0 is invariant of the dynamics, while J −2 q k , q k = 0 is not. If J −2 q k+1 , q k+1 = 0, by definition the flow stops. In this sense, in the codomain of Φ we should take the manifold defined without the assumption Q, J −2 Q = 0.

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

“…Similar discrete systems on pseudo-spheres and light-like cones in pseudo-Euclidean spaces are studied in [32]. Another discretization of the Neumann system as an even order Jacobi-Mumford systems, which has different first integrals in comparison with the classical case, was obtained by Ragnisco [52].…”

“…where, as above, the multiplier Γ is given by (6.2). In particular, the correspondence B r is symplectic (e.g., see [32,44,59]).…”