Degenerate integrable systems on the plane with a cubic integral of motion

We propose a classification of the known two-dimensional Hamiltonian systems of the natural form possessing an additional integral of motion that is cubic in the momenta. For degenerate systems of the Stäckel type, the additional cubic integral has the form of a “generalized angular momentum”. This allows constructing n-dimensional degenerate systems of the Stäckel type with additional cubic integrals of motion.     

On the bi-Hamiltonian structure of Bogoyavlensky system on <Emphasis Type="Italic">so</Emphasis>(4)

We discuss bi-Hamiltonian structure for the Bogoyavlensky system on so(4) with an additional integral of fourth order in momenta. An explicit procedure to find the variables of separation and the separation relations is considered in detail.     

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.     

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.     

A topological classification of integrable geodesic flows on the two-dimensional sphere with an additional integral quadratic in the momenta

Summary The analytic expression for a Riemannian metric on a 2-sphere, having integrable geodesic flow with an additional integral quadratic in momenta, is given in [Ko1]. We give the topological classification, up to topological equivalence of Liouville foliations, of all such metrics. The classification is computable, and the formula for calculating the complexity of the flow is straightforward. We prove Fomenko's conjecture that, from the point of view of complexity, the integrable geodesic flows with an additional integral linear or quadratic in momenta exhaust “almost all” integrable geodesic flows on the 2-dimensional sphere.     

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&nbsp;additional high degree first integral polynomial in momenta and is...     

Superintegrable models on Riemannian surfaces of revolution with integrals of any integer degree (I)

We present a family of superintegrable (SI) systems which live on a Riemannian surface of revolution and which exhibit one linear integral and two integrals of any integer degree larger or equal to 2 in the momenta. When this degree is 2, one recovers a metric due to Koenigs.The local structure of these systems is under control of a linear ordinary differential equation of order n which is homogeneous for even integrals and weakly inhomogeneous for odd integrals. The form of the integrals is explicitly given in the so-called “simple” case (see Definition 2). Some globally defined examples are worked out which live either in H² or in R².     

On the absence of an additional meromorphic first integral in the Riemann problem on the motion of a homogeneous liquid ellipsoid

We prove the absence of an additional meromorphic first integral in the Riemann problem on the motion of a homogeneous liquid ellipsoid with zero angular and vortex momenta in the case of zero self-gravitation.     

The construction of Poincaré-Chetayevand Smale bifurcation diagrams for conservative non-holonomic systems with symmetry

The problem of the existence of first integrals which are linear functions of the generalized velocities (momenta and quasi-velocities) is discussed for conservative non-holonomic Chaplygin systems with symmetry, as well as methods for investigating the existence, stability, and bifurcation of the steady motions of such systems. These methods are based on the classical methods of Routh-Salvadori, Poincaré-Chetayev, and Smale, but unlike the latter they do not require a knowledge of the explicit form of the linear integrals. The general conclusions are illustrated by the example of the problem of an ellipsoid of revolution moving on an absolutely rough horizontal surface. It is shown how in this case numerical techniques can be used to construct the Poincaré-Chetayev diagram — a surface in the space of generalized coordinates and constants of linear first integrals corresponding to motions in which the velocities of the non-cyclic coordinates vanish, while those of the cyclic coordinates are constant, and the Smale diagram — a surface in the space of constants of linear first integrals and the energy integral corresponding to these motions.     

一类Zakharov-Kuznestov方程解的正则性

In this paper, we consider the regularity of solution in S for Zakharov-Kuznestov equation in Hs(s > 2). Meanwhile, by method of undetermined coefficient we prove that there don't exist and conservative integral include 2 order of higher order derived functions.     

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.     

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.     

Techniques for searching first integrals by lie group and application to gyroscope system

In the paper, the methods of finding first integrals of an autonomous system using one-parameter Lie groups are discussed. A class of nontrivial one-parameter Lie groups admitted by the classical gyroscope system is found, and based on the properties of first integral determined by the one-parameter Lie group, the fourth first integral of the gyroscope system in Euler case, Lagrange case and Kovalevskaya case can be obtained in a uniform idea. An error on the fourth first integral in general Kovalevskaya case (A = B = 2C, ZG =0), which appeared in literature is found and corrected.     

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.     

Bäcklund transformations for the nonholonomic Veselova system

We present auto and hetero Bäcklund transformations of the nonholonomic Veselova system using standard divisor arithmetic on the hyperelliptic curve of genus two. As a by-product one gets two natural integrable systems on the cotangent bundle to the unit two-dimensional sphere whose additional integrals of motion are polynomials in the momenta of fourth order.     

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.     

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.     

Orthogonality Relations for a Stationary Flow of an Ideal Fluid

For a real solution (u, p) to the Euler stationary equations for an ideal fluid, we derive an infinite series of the orthogonality relations that equate some linear combinations of mth degree integral momenta of the functions u_iu_j and p to zero (m = 0, 1,... ). In particular, the zeroth degree orthogonality relations state that the components ui of the velocity field are L²-orthogonal to each other and have coincident L²-norms. Orthogonality relations of degree m are valid for a solution belonging to a weighted Sobolev space with the weight depending on m.     

一类核密度含高阶奇性Cauchy型积分的边值定理

本文推广「１」，「６」中的结果，讨论了一类开口弧核密度含高阶奇且情形更一般的Ｃａｕｃｈｙ型积分的边值定理，积分号下求导及Ｈ连续性。     

Integral representation of slowly growing equidistant splines

In this paper we consider polynomial splines S(x) with equidistant nodes which may grow as O (|x|^s). We present an integral representation of such splines with a distribution kernel. This representation is related to the Fourier integral of slowly growing functions. The part of the Fourier exponentials herewith play the so called exponential splines by Schoenberg. The integral representation provides a flexible tool for dealing with the growing equidistant splines. First, it allows us to construct a rich library of splines possessing the property that translations of any such spline form a basis of corresponding spline space. It is shown that any such spline is associated with a dual spline whose translations form a biorthogonal basis. As examples we present solutions of the problems of projection of a growing function onto spline spaces and of spline interpolation of a growing function. We derive formulas for approximate evaluation of splines projecting a function onto the spline space and establish therewith exact estimations of the approximation errors.