Algorithms for the construction of an optimal cover for sets in three-dimensional Euclidean space

For a geodesic (or magnetic geodesic) flow, the problem of the existence of an additional (independent of the energy) first integral that is polynomial in momenta is studied. The relation of this problem to the existence of nontrivial solutions of stationary dispersionless limits of two-dimensional soliton equations is demonstrated. The nonexistence of an additional quadratic first integral is established for certain classes of magnetic geodesic flows.     

Some Remarks on High Degree Polynomial Integrals of the Magnetic Geodesic Flow on the Two-Dimensional Torus

Siberian Mathematical Journal - We study the magnetic geodesic flow on the two-dimensional torus which admits an additional high degree first integral polynomial in momenta and is...     

Characterization of Geodesic Flows on T 2 with and without Positive Topological Entropy

In the present work we consider the behavior of the geodesic flow on the unit tangent bundle of the 2-torus T ² for an arbitrary Riemannian metric. A natural non-negative quantity which measures the complexity of the geodesic flow is the topological entropy. In particular, positive topological entropy implies chaotic behavior on an invariant set in the phase space of positive Hausdorff-dimension (horseshoe). We show that in the case of zero topological entropy the flow has properties similar to integrable systems. In particular, there exists a non-trivial continuous constant of motion which measures the direction of geodesics lifted onto the universal covering \mathbbR²{\mathbb{R}^{2}} . Furthermore, those geodesics travel in strips bounded by Euclidean lines. Moreover, we derive necessary and sufficient conditions for vanishing topological entropy involving intersection properties of single geodesics on T ².     

On the generalized Chaplygin system

We consider two polynomial bi—Harnilt0nian structures for the generalized integrable Chaplygin system on the sphere S ² with an additional integral of fourth order in momenta. An explicit procedure for finding variables of separation, separation relations, and transformation of the corresponding algebraic curves of genus two is considered in detail. Bibliography: 21 titles.     

On the extendability of Noether’s integrals for orbifolds of constant negative curvature

This paper is concerned with the problem of the integrable behavior of geodesics on homogeneous factors of the Lobachevsky plane with respect to Fuchsian groups (orbifolds). Locally the geodesic equations admit three independent Noether integrals linear in velocities (energy is a quadratic form of these integrals). However, when passing along closed cycles the Noether integrals undergo a linear substitution. Thus, the problem of integrability reduces to the search for functions that are invariant under these substitutions. If a Fuchsian group is Abelian, then there is a first integral linear in the velocity (and independent of the energy integral). Conversely, if a Fuchsian group contains noncommuting hyperbolic or parabolic elements, then the geodesic flow does not admit additional integrals in the form of a rational function of Noether integrals. We stress that this result holds also for noncompact orbifolds, when there is no ergodicity of the geodesic flow (since nonrecurrent geodesics can form a set of positive measure).     

Integrable discretization and deformation of the nonholonomic Chaplygin ball

The rolling of a dynamically balanced ball on a horizontal rough table without slipping was described by Chaplygin using Abel quadratures. We discuss integrable discretizations and deformations of this nonholonomic system using the same Abel quadratures. As a by-product one gets a new geodesic flow on the unit two-dimensional sphere whose additional integrals of motion are polynomials in the momenta of fourth order.     

Integrable Geodesic Flows on Surfaces

We propose a new condition à{{\aleph}} which enables us to get new results on integrable geodesic flows on closed surfaces. This paper has two parts. In the first, we strengthen Kozlov’s theorem on non-integrability on surfaces of higher genus. In the second, we study integrable geodesic flows on a 2-torus. Our main result for a 2-torus describes the phase portraits of integrable flows. We prove that they are essentially standard outside what we call separatrix chains. The complement to the union of the separatrix chains is C ⁰-foliated by invariant sections of the bundle.     

Liouville classification of integrable geodesic flows on a torus of revolution in a potential field

A Liouville classification of integrable Hamiltonian systems being geodesic flows on a twodimensional torus of revolution in an invariant potential field is obtained in the case of linear integral. This classification is obtained using the Fomenko–Zieschang invariant (so called marked molecules) of the systems under consideration. All types of bifurcation curves are described. A classification of singularities of the system solutions is also obtained.     

An integrable system related to the spherical top and the Toda chain

We study the integrable motion over the sphere S² in the potential V=(x₁x₂x₃)^−2/3 possessing an additional integral of motion that is cubic in the momenta. We construct the Lax representation without a spectral parameter and consider the relation to the three-particle Toda chain. Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 124, No. 2, pp. 310–322, August, 2000.     

Quasi-Stäckel systems and two-dimensional Schrödinger equations in an electromagnetic field

We obtain the complete classification of two-dimensional Schrödinger equations in an electromagnetic field with an additional integral quadratic in momenta. For this, we use a Kovalevskaya-type change of variables and reduce the Hamiltonians to a quasi-Stäckel form. In that form, we perform the classification in the Painlevé sense and then return to the original variables.     

Two integrable systems with integrals of motion of degree four

We discuss the possibility of using second-order Killing tensors to construct Liouville-integrable Hamiltonian systems that are not Nijenhuis integrable. As an example, we consider two Killing tensors with a nonzero Haantjes torsion that satisfy weaker geometric conditions and also three-dimensional systems corresponding to them that are integrable in Euclidean space and have two quadratic integrals of motion and one fourth-order integral in momenta.     

New Families of Conservative Systems on S2 Possessing an Integral of Fourth Degree in Momenta

There is a well-known example of an integrable conservative system on S2, the case of Kovalevskaya in the dynamics of a rigid body, possessing an integral of fourth degree in momenta. In this paper we propose new families of examples of conservative systems on S2 possessing an integral of fourth degree in momenta.     

Quadratic Integrals of Motion for Systems of Identical Particles: The Quantum Case

We consider the quantum dynamical systems of identical particles admitting an additional integral quadratic in momenta and find that an appropriate ordering procedure exists that allows converting the classical integrals into their quantum counterparts. We discuss the relation to the separation of variables in the Schrodinger equation. __________ Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 144, No. 2, pp. 290–294, August, 2005.     

An Ellipsoidal Billiard with a Quadratic Potential

There exists an infinite hierarchy of integrable generalizations of the geodesic flow on an n-dimensional ellipsoid. These generalizations describe the motion of a point in the force fields of certain polynomial potentials. In the limit as one of semiaxes of the ellipsoid tends to zero, one obtains integrable mappings corresponding to billiards with polynomial potentials inside an (n-1)-dimensional ellipsoid.In this paper, for the first time we give explicit expressions for the ellipsoidal billiard with a quadratic (Hooke) potential, its representation in Lax form, and a theta function solution. We also indicate the generating function of the restriction of the potential billiard map to a level set of an energy type integral. The method we use to obtain theta function solutions is different from those applied earlier and is based on the calculation of limit values of meromorphic functions on generalized Jacobians.     

Integrable magnetic geodesic flows on lie groups

On Lie group manifolds, we consider right-invariant magnetic geodesic flows associated with 2-cocycles of the corresponding Lie algebras. We investigate the algebra of the integrals of motion of magnetic geodesic flows and also formulate a necessary and sufficient condition for their integrability in quadratures, giving the canonical forms of 2-cocycles for all four-dimensional Lie algebras and selecting integrable cases. __________ Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 156, No. 2, pp. 189–206, August, 2008.     

Projectively Equivalent Riemannian Spaces as Quasi-bi-Hamiltonian Systems

The class of Riemannian spaces admitting projectively, or geodesically, equivalent metrics is very closely related to a certain class of spaces for which the Hamilton–Jacobi equation for geodesics is separable. This fact is established, and its consequences explored, by showing that when a Riemannian space has a projectively equivalent metric its geodesic flow is a quasi-bi-Hamiltonian system. The existence of involutive first integrals of the geodesic flow, quadratic in the momenta, follows by a standard type of argument. When these integrals are independent they generate a Stäckel system.     

THE SCENERY FLOW FOR GEOMETRIC STRUCTURES ON THE TORUS： THE LINEAR SETTING

50. IntroductionWe begin by recalling some wellknown relationshiPs. First, ther is the one-to-one corre-spondence between closed orbits of the g6odesic fiow on the modular surfaCe and conjugacyclasses of hyperbolic toral automorphisms. (This can be seen directly from the definitions(see Remaxk 1.3 in 51 below).) Secondly one knows that it is possible to code this geodesicflow using coatinued fractions and via circle rotations (cf [9, 42, 2, 7J). Thirdly, there is astrong relation between hyp…     

Geodesic equivalence of metrics as a particular case of integrability of geodesic flows

We consider the recently found connection between geodesically equivalent metrics and integrable geodesic flows. If two different metrics on a manifold have the same geodesics, then the geodesic flows of these metrics admit sufficiently many integrals (of a special form) in involution, and vice versa. The quantum version of this result is also true: if two metrics on one manifold have the same geodesics, then the Beltrami Laplace operator Δ for each metric admits sufficiently many linear differential operators communiting with Δ. This implies that the topology of a manifold with two different metrics with the same geodesics must be sufficiently simple. We also have that the nonproportionality of the metrics at a point implies the nonproportionality of the metrics at almost all points. In memory of Mikhail Vladimirovich Saveliev Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 123, No. 2, pp. 285–293, May, 2000.     

Unique ergodicity of horospheric foliations revisited

With each rational function on the Riemann sphere, Lyubich–Minsky construction (1997) associates an abstract topological space called the quotient hyperbolic lamination. The latter space carries the so-called vertical geodesic flow with Anosov property. Its unstable foliation is what we call the quotient horospheric lamination. We consider the case of hyperbolic rational function, and more generally, functions postcritically finite on the Julia set without parabolics, that do not belong to the following list of exceptions: powers, Chebyshev polynomials and Latt‘es examples. In this case the quotient horospheric lamination is known to be minimal, while restricted to the union of nonisolated hyperbolic leaves (Glutsyuk, 2007). In the present paper we prove its unique ergodicity. To this end, we introduce the so-called transversely contracting flows and homeomorphisms (on abstract compact metrizable topological spaces), which include the vertical geodesic flows under consideration and the usual Anosov flows and diffeomorphisms. We prove a version of our unique ergodicity result for the transversely contracting flows and homeomorphisms. Particular cases for Anosov flows and diffeomorphisms are given by classical results by Bowen, Marcus, Furstenberg, Margulis, et al. We give a new and purely geometric proof, which seems to be simpler than the classical ones (which use either Markov partitions, K-property, or harmonic analysis).     

On a class of integrable systems with a quartic first integral

We generalize, to some extent, the results on integrable geodesic flows on two dimensional manifolds with a quartic first integral in the framework laid down by Selivanova and Hadeler. The local structure is first determined by a direct integration of the differential system which expresses the conservation of the quartic observable and is seen to involve a finite number of parameters. The global structure is studied in some detail and leads to a class of models on the manifolds {ie394-1}², ?² or ?². As special cases we recover Kovalevskaya’s integrable system and a generalization of it due to Goryachev.