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

Jovanović, Božidar

doi:10.1007/s00205-013-0638-4

Cited by 11 publications

(16 citation statements)

References 48 publications

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“…The first statement is an analog of Theorems 5.1, 5.2 for the the Jacobi-Rosochatius problem [20] and Theorem 4.1 for geodesic flows on quadrics in pseudo-Euclidean spaces [23], where the Dirac construction is applied for the constraints…”

Section: Symmetric Quadricsmentioning

confidence: 94%

See 1 more Smart Citation

Virtual billiards in pseudo–euclidean spaces: Discrete hamiltonian and contact integrability

Jovanović¹,

Jovanović²

2017

Discrete &Amp; Continuous Dynamical Systems - A

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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 integrability. As a starting point in this direction, we consider virtual billiard dynamics within quadrics in pseudo-Euclidean spaces. In contrast to the usual billiards, the incoming velocity and the velocity after the billiard reflection can be at opposite sides of the tangent plane at the reflection point. In the symmetric case we prove noncommutative integrability of the system and give a geometrical interpretation of integrals, an analog of the classical Chasles and Poncelet theorems and we show that the virtual billiard dynamics provides a natural framework in the study of billiards within quadrics in projective spaces, in particular of billiards within ellipsoids on the sphere S n−1 and the Lobachevsky space H n−1 .1991 Mathematics Subject Classification. 70H06, 37J35, 37J55, 51M05.

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Section: Symmetric Quadricsmentioning

confidence: 94%

“…Note that the virtual billiard dynamics on Q n−1 can have both virtual and real reflections. Motivated by the Lax reprezentation for elliptical billiards with the Hooke's potential (Fedorov [16], see also [20,32]), we proved in [23] that the trajectories (x j , y j ) of (3), (4) outside the singular set (5) satisfy the matrix equation…”

Section: Introductionmentioning

confidence: 99%

Virtual billiards in pseudo–euclidean spaces: Discrete hamiltonian and contact integrability

Jovanović¹,

Jovanović²

2017

Discrete &Amp; Continuous Dynamical Systems - A

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“…Then the geodesic flow of (5.8) and the system (5.12) are SO(|I 0 |)×· · ·× SO(|I r |)-symmetric with Noether's integrals (5.21) and (5.22), respectively. For ǫ = ±1 they are completely integrable in a non-commutative sense by means of Noether's integrals and commuting integrals that are certain limits of integrals of a non-symmetric case (e.g, see [18,30], where a detail analysis for natural mechanical systems on a symmetric ellipsoid is given). The corresponding Hamiltonian flows on the cotangent bundle of a sphere S n−1 are generically quasi-periodic over invariant isotropic tori of dimension N = r + ♯{I i ||I i | ≥ 2}.…”

Section: Noncommutative Integrability Of a Symmetric Case (ǫ = ±1mentioning

confidence: 99%

Nonholonomic connections, time reparametrizations, and integrability of the rolling ball over a sphere

Gajić

Jovanović

2019

Nonlinearity

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We study a time reparametrisation of the Newton type equations on Riemannian manifolds slightly modifying the Chaplygin multiplier method, allowing us to consider the Chaplygin method and the Maupertuis principle within a unified framework. As an example, the reduced nonholonomic problem of rolling without slipping and twisting of an n-dimensional balanced ball over a fixed sphere is considered. For a special inertia operator (depending on n parameters) we prove complete integrability when the radius of the ball is twice the radius of the sphere. In the case of SO(l) × SO(n − l) symmetry, noncommutative integrability for any ratio of the radii is established.

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“…Example 2.1. As an example of a system with symmetry, consider the billiard within ellipsoid [13] we studied the reduction of symmetries of the given billiard with additional Hook's potential). In the complex notation we have…”

Section: The Skew Hodograph Mapping and Quadratic Generating Functionsmentioning

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

Cited by 11 publications

References 48 publications

Virtual billiards in pseudo–euclidean spaces: Discrete hamiltonian and contact integrability

Virtual billiards in pseudo–euclidean spaces: Discrete hamiltonian and contact integrability

Nonholonomic connections, time reparametrizations, and integrability of the rolling ball over a sphere

Heisenberg Model in Pseudo-Euclidean Spaces II

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