53 research outputs found

### Multiplication of solutions for linear overdetermined systems of partial differential equations

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

Systems of Newton equations of the form $\ddot{q}=-{1/2}A^{-1}(q)\nabla k$
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

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 $\theta(t)$ 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 $r$-matrices.

We study a quite general family of dynamical $r$-matrices for an auxiliary
loop algebra ${\cal L}({su(2)})$ related to restricted flows for equations of
the KdV type. This underlying $r$-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

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

### Linear $r$-Matrix Algebra for Systems Separable\\ in Parabolic Coordinates

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 $2\times 2$ matrices for the whole hierarchy, we construct
the associated linear $r$-matrix algebra with the $r$-matrix dependent on the
dynamical variables. A dynamical Yang-Baxter equation is discussed.Comment: 10 pages, LaTeX. Submitted to Phys.Lett.

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