On Reciprocal Equivalence of Stäckel Systems

In this paper we investigate Stäckel transforms between different classes of parameter‐dependent Stäckel separable systems of the same dimension. We show that the set of all Stäckel systems of the same dimension splits to equivalence classes so that all members within the same class can be connected by a single Stäckel transform. We also give an explicit formula relating solutions of two Stäckel‐related systems. These results show in particular that any two geodesic Stäckel systems are Stäckel equivalent in the sense that it is possible to transform one into another by a single Stäckel transform. We also simplify proofs of some known statements about multiparameter Stäckel transform.     

On maximally superintegrable systems

Locally any completely integrable system is maximally superintegrable system since we have the necessary number of the action-angle variables. The main problem is the construction of the single-valued additional integrals of motion on the whole phase space by using these multi-valued action-angle variables. Some constructions of the additional integrals of motion for the Stäckel systems and for the integrable systems related with two different quadratic r-matrix algebras are discussed. Among these system there are the open Heisenberg magnet and the open Toda lattices associated with the different root systems.     

Stäckel transform of Lax equations

We construct a map between Lax equations for pairs of Liouville integrable Hamiltonian systems related by a multiparameter Stäckel transform. Using this map, we construct Lax representation for a wide class of separable systems by applying the multiparameter Stäckel transform to Lax equations of suitably chosen systems from a seed class. For a given separable system, we obtain in this way a set of nonequivalent Lax equations parameterized by an arbitrary function of the spectral parameter, as it is in the case of a related seed system.     

Leonard Euler: Addition theorems and superintegrable systems

We consider the Euler approach to constructing to investigating of the superintegrable systems related to the addition theorems. As an example we reconstruct Drach systems and get some new two-dimensional superintegrable Stäckel systems.     

Homogeneous Stäckel-type systems

A class of Hamiltonian dynamic systems integrated by the variable separation method is considered. The integration for this class is the inversion of an Abel mapping on hyperelliptic curves. We prove that the derivative of the Abel mapping is the Stäckel matrix, which determines a diagonal Riemannian metric and curvilinear orthogonal coordinate systems in a flat space. Lax representations with the spectral parameter are constructed. The corresponding classicalr-matrices are dynamic. It is shown how the class of pointwise canonical transformations can be naturally generalized using the Abel integral reduction theory.     

Noncanonical time transformations relating finite-dimensional integrable systems

We consider dual Stäckel schemes related to each other by a noncanonical transformation of the time variable. We prove that this duality of different integrable systems arises from the multivaluedness of the Abel mapping. We construct the Lax matrices and the r-matrix algebras for some integrable systems on a plane. The integrable deformations of the Kepler problem and the Holt-type systems are considered in detail.     

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.     

Bi‐Presymplectic Representation of Liouville Integrable Systems and Related Separability Theory

Bi‐presymplectic chains of one‐forms of arbitrary co‐rank are considered. The conditions in which such chains represent some Liouville integrable systems and the conditions in which there exist related bi‐Hamiltonian chains of vector fields are presented. To derived the construction of bi‐presymplectic chains, the notions of dual Poisson‐presymplectic pair, d‐compatibility of presymplectic forms and d‐compatibility of Poisson bivectors is used. The completely algorithmic construction of separation coordinates is demonstrated. It is also proved that Stäckel separable systems have bi‐inverse‐Hamiltonian representation, i.e., are represented by bi‐presymplectic chains of closed one‐forms. The co‐rank of related structures depends on the explicit form of separation relations.     

Canonical transformations of the extended phase space and integrable systems

We investigate the explicit construction of a canonical transformation of the time variable and the Hamiltonian whereby a given completely integrable system is mapped into another integrable system. The change of time induces a transformation of the equations of motion and of their solutions, the integrals of motion, the methods of separation of variables, the Lax matrices, and the correspondingr-matrices. For several specific families of integrable systems (Toda chains, Holt systems, and Stäckel-type systems), we construct canonical transformations of time in the extended phase space that preserve the integrability property.     

Bäcklund transformations for the Jacobi system on an ellipsoid

We consider analogues of auto- and hetero-Bäcklund transformations for the Jacobi system on a threeaxis ellipsoid. Using the results in a Weierstrass paper, where the change of times reduces integrating the equations of motion to inverting the Abel mapping, we construct the differential Abel equations and auto-Bäcklund transformations preserving the Poisson bracket with respect to which the equations of motion written in the Weierstrass form are Hamiltonian. Transforming this bracket to the canonical form, we can construct a new integrable system on the ellipsoid with a Hamiltonian of the natural form and with a fourth-degree integral of motion in momenta.     

From Jacobi problem of separation of variables to theory of quasipotential Newton equations

Our solution to the Jacobi problem of finding separation variables for natural Hamiltonian systems H = ½p ² + V(q) is explained in the first part of this review. It has a form of an effective criterion that for any given potential V(q) tells whether there exist suitable separation coordinates x(q) and how to find these coordinates, so that the Hamilton-Jacobi equation of the transformed Hamiltonian is separable. The main reason for existence of such criterion is the fact that for separable potentials V(q) all integrals of motion depend quadratically on momenta and that all orthogonal separation coordinates stem from the generalized elliptic coordinates. This criterion is directly applicable to the problem of separating multidimensional stationary Schrödinger equation of quantum mechanics. Second part of this work provides a summary of theory of quasipotential, cofactor pair Newton equations $ \ddot q $ = M(q) admitting n quadratic integrals of motion. This theory is a natural generalization of theory of separable potential systems $ \ddot q $ = ??(q). The cofactor pair Newton equations admit a Hamilton-Poisson structure in an extended 2n + 1 dimensional phase space and are integrable by embedding into a Liouville integrable system. Two characterizations of these systems are given: one through a Poisson pencil and another one through a set of Fundamental Equations. For a generic cofactor pair system separation variables have been found and such system have been shown to be equivalent to a Stäckel separable Hamiltonian system. The theory is illustrated by examples of driven and triangular Newton equations.     

On a ogawa–type integral with application to the fractional brownian motion

We construct a deterministic Ogawa–type integral with respect to a continuous function that, in particular, can be a trajectory of the Fractional Brownian motion. This integral is related with the Stratonovich integral and with the integrals introduced by Ciesielski et altri and Zähle. We give a sufficient condition for the integrability of a function in this sense, that does not imply its continuity. Under this sufficient condition, we obtain a Besov regularity property of the indefinite integral. We also study the stochastic Ogawa integral for stochastic processes when integrate with respect to the Fractional Brownian motion of Hurst parameter H ∈ (1/2, 1)     

First integrals and families of symmetric periodic motions of a reversible mechanical system

A reversible mechanical system which allows of first integrals is studied. It is established that, for symmetric motions, the constants of the asymmetric integrals are equal to zero. The form of the integrals of a reversible linear periodic system corresponding to zero characteristic exponents and the structure of the corresponding Jordan Boxes are investigated. A theorem on the non-existence of an additional first integral and a theorem on the structural stabilities of having a symmetric periodic motion (SPM) are proved for a system with m symmetric and k asymmetric integrals. The dependence of the period of a SPM on the constants of the integrals is investigated. Results of the oscillations of a quasilinear system in degenerate cases are presented. Degeneracy and the principal resonance: bifurcation with the disappearance of the SPM and the birth of two asymmetric cycles, are investigated. A heavy rigid body with a single fixed point is studied as the application of the results obtained. The Euler-Poisson equations are used. In the general case, the energy integral and the geometric integral are symmetric while the angular momentum integral turns out to be asymmetric. In the special case, when the centre of gravity of the body lies in the principal plane of the ellipsoid of inertia, all three classical integrals become symmetric. It is ascertained here that any SPM of a body contains four zero characteristic exponents, of which two are simple and two form a Jordan Box. In typical situation, the remaining two characteristic exponents are not equal to zero. All of the above enables one to speak of an SPM belonging to a two-parameter family and the absence of an additional first integral. It is established that a body also executes a pendulum motion in the case when the centre of gravity is close to the principal plane of the ellipsoid of inertia.     

Inverse problem for a nonlinear Benney–Luke type integro-differential equations with degenerate kernel

We consider the problem on the unique solvability of the inverse problem for a nonlinear partial Benney–Luke type integro-differential equation of the fourth order with a degenerate kernel. We modify the degenerate kernelmethod which has been designed for Fredholm integral equations of the second kind to apply to the case of the above-mentioned equation. We exploit the Fouriermethod of separation of variables. By means of designations, the Benney–Luke type integro-differential equation is reduced to a system of algebraic equations. Using an additional condition, we obtain the countable system of nonlinear integral equations with respect to the main unknown function. We employ the method of successive approximations together with the contraction mapping principle. Finally, the restore function is defined.     

Spatial impedance control of two cooperative industrial manipulators

Nowadays robotic manipulators are considered to perform a wide range of tasks. Since robotic systems consisting of multiple manipulators are capable of handling a variety of tasks that cannot be executed by single manipulators, cooperation of multiple manipulators is becoming increasingly interesting in particular for industrial applications. The interaction with other manipulators, respectively the environment, requires an extension of the conventional position control in order to achieve a desired compliance, and thus to limit the interaction wrench so to avoid damaging the involved objects. In this contribution the cooperation of two industrial manipulators, a Stäubli RX130L (6-DOF) and a Stäubli TX90L mounted on a linear axis (constituting a redundant 7-DOF manipulator), is addressed applying an impedance control scheme. The manipulation task is to grasp an object with both manipulators and to follow a prescribed trajectory by simultaneously limiting the contact forces between the manipulators and the object and ensuring a compliant behavior towards the environment. (© 2016 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Coupled vortex equations and moduli: deformation theoretic approach and Kähler geometry

We investigate differential geometric aspects of moduli spaces parametrizing solutions of coupled vortex equations over a compact Kähler manifold X. These solutions are known to be related to polystable triples via a Kobayashi–Hitchin type correspondence. Using a characterization of infinitesimal deformations in terms of the cohomology of a certain elliptic double complex, we construct a Hermitian structure on these moduli spaces. This Hermitian structure is proved to be Kähler. The proof involves establishing a fiber integral formula for the Hermitian form. We compute the curvature tensor of this Kähler form. When X is a Riemann surface, the holomorphic bisectional curvature turns out to be semi-positive. It is shown that in the case where X is a smooth complex projective variety, the Kähler form is the Chern form of a Quillen metric on a certain determinant line bundle.     

Alternative derivations for the Poisson integral formula

Poisson integral formula is revisited. The kernel in the Poisson integral formula can be derived in a series form through the direct BEM free of the concept of image point by using the null-field integral equation in conjunction with the degenerate kernels. The degenerate kernels for the closed-form Green's function and the series form of Poisson integral formula are also derived. Two and three-dimensional cases are considered. Also, interior and exterior problems are both solved. Even though the image concept is required, the location of image point can be determined straightforward through the degenerate kernels instead of the method of reciprocal radii.     

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.     

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.     

Mixed problem for pseudoparabolic integro-differential equation with degenerate kernel

A mixed problem for a certain nonlinear third-order intregro-differential equation of the pseudoparabolic type with a degenerate kernel is considered. The method of degenerate kernel is essentially used and developed and the Fourier method of variable separation is employed for this equation. A system of countable systems of algebraic equations is first obtained; after it is solved, a countable system of nonlinear integral equations is derived. The method of sequential approximations is used to prove the theorem on the unique solvability of the mixed problem.