62 research outputs found

    Multiplication of solutions for linear overdetermined systems of partial differential equations

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    A large family of linear, usually overdetermined, systems of partial differential equations that admit a multiplication of solutions, i.e, a bi-linear and commutative mapping on the solution space, is studied. This family of PDE's contains the Cauchy-Riemann equations and the cofactor pair systems, included as special cases. The multiplication provides a method for generating, in a pure algebraic way, large classes of non-trivial solutions that can be constructed by forming convergent power series of trivial solutions.Comment: 27 page

    Quasi-Lagrangian Systems of Newton Equations

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    Systems of Newton equations of the form q¨=1/2A1(q)k\ddot{q}=-{1/2}A^{-1}(q)\nabla k with an integral of motion quadratic in velocities are studied. These equations generalize the potential case (when A=I, the identity matrix) and they admit a curious quasi-Lagrangian formulation which differs from the standard Lagrange equations by the plus sign between terms. A theory of such quasi-Lagrangian Newton (qLN) systems having two functionally independent integrals of motion is developed with focus on two-dimensional systems. Such systems admit a bi-Hamiltonian formulation and are proved to be completely integrable by embedding into five-dimensional integrable systems. They are characterized by a linear, second-order PDE which we call the fundamental equation. Fundamental equations are classified through linear pencils of matrices associated with qLN systems. The theory is illustrated by two classes of systems: separable potential systems and driven systems. New separation variables for driven systems are found. These variables are based on sets of non-confocal conics. An effective criterion for existence of a qLN formulation of a given system is formulated and applied to dynamical systems of the Henon-Heiles type.Comment: 50 pages including 9 figures. Uses epsfig package. To appear in J. Math. Phy

    Dynamics of the Tippe Top -- properties of numerical solutions versus the dynamical equations

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    We study the relationship between numerical solutions for inverting Tippe Top and the structure of the dynamical equations. The numerical solutions confirm oscillatory behaviour of the inclination angle θ(t)\theta(t) for the symmetry axis of the Tippe Top. They also reveal further fine features of the dynamics of inverting solutions defining the time of inversion. These features are partially understood on the basis of the underlying dynamical equations

    Stationary problems for equation of the KdV type and dynamical rr-matrices.

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    We study a quite general family of dynamical rr-matrices for an auxiliary loop algebra L(su(2)){\cal L}({su(2)}) related to restricted flows for equations of the KdV type. This underlying rr-matrix structure allows to reconstruct Lax representations and to find variables of separation for a wide set of the integrable natural Hamiltonian systems. As an example, we discuss the Henon-Heiles system and a quartic system of two degrees of freedom in detail.Comment: 25pp, LaTe

    Natural coordinates for a class of Benenti systems

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    We present explicit formulas for the coordinates in which the Hamiltonians of the Benenti systems with flat metrics take natural form and the metrics in question are represented by constant diagonal matrices.Comment: LaTeX 2e, 8 p., no figures; extended version with enlarged bibliograph

    Algebraic Integrability Conditions for Killing Tensors on Constant Sectional Curvature Manifolds

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    We use an isomorphism between the space of valence two Killing tensors on an n-dimensional constant sectional curvature manifold and the irreducible GL(n+1)-representation space of algebraic curvature tensors in order to translate the Nijenhuis integrability conditions for a Killing tensor into purely algebraic integrability conditions for the corresponding algebraic curvature tensor, resulting in two simple algebraic equations of degree two and three. As a first application of this we construct a new family of integrable Killing tensors.Comment: 34 pages, no figure

    Linear rr-Matrix Algebra for Systems Separable\\ in Parabolic Coordinates

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    We consider a hierarchy of many particle systems on the line with polynomial potentials separable in parabolic coordinates. Using the Lax representation, written in terms of 2×22\times 2 matrices for the whole hierarchy, we construct the associated linear rr-matrix algebra with the rr-matrix dependent on the dynamical variables. A dynamical Yang-Baxter equation is discussed.Comment: 10 pages, LaTeX. Submitted to Phys.Lett.