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

This paper uncovers a large class of left-invariant sub-Riemannian systems on Lie groups that admit explicit solutions with certain properties, and provides geometric origins for a class of important curves on Stiefel manifolds, called quasi-geodesics, that project on Grassmann manifolds as Riemannian geodesics. We show that quasi-geodesics are the projections of sub-Riemannian geodesics generated by certain leftinvariant distributions on Lie groups that act transitively on each Stiefel manifold St n k (V ). This result is valid not only for the real Stiefel manifolds in V = R n , but also for the Stiefels in the Hermitian space V = C n and the quaternion space V = H n .

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“…Their projections on St n k (V ) are given by These geodesics are called canonical in [7] and normal in [8].…”

Section: Sub-riemannian Geodesicsmentioning

confidence: 99%

Extremal Curves on Stiefel and Grassmann Manifolds

2019

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“…A "big" n × n matrix representation and integration of equations (3.3) in the signature (n − 1, 1), i.e, of the Neumann system in the Lobachevsky space is given by Veselov (see Appendix B, [27]). A generalization of the Neumann system to the Stiefel varieties, as well as its integrable discretization, is given in [10] and [11], respectively.…”

Section: The Skew Hodograph Mapping and Quadratic Generating Functionsmentioning

confidence: 99%

“…For the Euclidean case it is proved by Moser (e.g., see [22]). The above proof is taken from [10], where it is given for the Neumann systems on Stiefel varieties.…”

Section: Geometric Interpretation Of the Integralsmentioning

confidence: 99%

Heisenberg Model in Pseudo-Euclidean Spaces II

Jovanović

2018

Regul. Chaot. Dyn.

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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.

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“…, Q ηn−1 for periodic billiard trajectories is derived by Dragović and Radnović, generalizing classical Cayley's condition for n = 2 [15,16]. The geometry of the lines common to the confocal quadrics is further studied in [30,16], while Chasles's-type theorems for several natural mechanical systems are given in [33,19,22].…”

Section: The Chasles and Poncelet Theoremsmentioning

confidence: 99%

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

Jovanović

2013

Arch Rational Mech Anal

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The Lax representations of the geodesic flow, the Jacobi-Rosochatius problem and its perturbations by means of separable polynomial potentials, on an ellipsoid are constructed. We prove complete integrability in the case of a generic symmetric ellipsoid and describe analogous systems on complex projective spaces. Also, we consider billiards within an ellipsoid under the influence of the Hook and Rosochatius potentials between the impacts. A geometric interpretation of the integrability analogous to the classical Chasles and Poncelet theorems is given.

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

Cited by 12 publications

References 75 publications

Extremal Curves on Stiefel and Grassmann Manifolds

Extremal Curves on Stiefel and Grassmann Manifolds

Heisenberg Model in Pseudo-Euclidean Spaces II

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

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