Volume 12, Number 2
Volume 12, Number 2, 2007
O'Neil K. A.
Relative Equilibrium and Collapse Configurations of Four Point Vortices
Abstract
Relative equilibrium configurations of point vortices in the plane can be related to a system of polynomial equations in the vortex positions and circulations. For systems of four vortices the solution set to this system is proved to be finite, so long as a number of polynomial expressions in the vortex circulations are nonzero, and the number of relative equilibrium configurations is thereby shown to have an upper bound of 56. A sharper upper bound is found for the special case of vanishing total circulation. The polynomial system is simple enough to allow the complete set of relative equilibrium configurations to be found numerically when the circulations are chosen appropriately. Collapse configurations of four vortices are also considered; while finiteness is not proved, the approach provides an effective computational method that yields all configurations with a given ratio of velocity to position.

Koiller J., Ehlers K. M.
Rubber Rolling over a Sphere
Abstract
"Rubber" coated bodies rolling over a surface satisfy a notwist condition in addition to the no slip condition satisfied by "marble" coated bodies [1]. Rubber rolling has an interesting differential geometric appeal because the geodesic curvatures of the curves on the surfaces at corresponding points are equal. The associated distribution in the 5 dimensional configuration space has 235 growth (these distributions were first studied by Cartan; he showed that the maximal symmetries occurs for rubber rolling of spheres with 3:1 diameters ratio and materialize the exceptional group $G_2$). The 235 nonholonomic geometries are classified in a companion paper [2] via Cartan's equivalence method [3]. Rubber rolling of a convex body over a sphere defines a generalized Chaplygin system [48] with $SO$(3) symmetry group, total space $Q = SO(3) \times S^2$ and base $S^2$, that can be reduced to an almost Hamiltonian system in $T^*S^2$ with a nonclosed 2form $\omega_{NH}$. In this paper we present some basic results on the spheresphere problem: a dynamically asymmetric but balanced sphere of radius $b$ (unequal moments of inertia $I_j$ but with center of gravity at the geometric center), rubber rolling over another sphere of radius $a$. In this example $\omega_{NH}$ is conformally symplectic [9]: the reduced system becomes Hamiltonian after a coordinate dependent change of time. In particular there is an invariant measure, whose density is the determinant of the reduced Legendre transform, to the power $p = 1/2 (b/a  1)$. Using spheroconical coordinates we verify the result by Borisov and Mamaev [10] that the system is integrable for $p = 1/2$ (ball over a plane). They have found another integrable case [11] corresponding to $p = 3/2$ (rolling ball with twice the radius of a fixed internal ball). Strikingly, a different set of spheroconical coordinates separates the Hamiltonian in this case. No other integrable cases with different $I_j$ are known.

Borisov A. V., Mamaev I. S.
Rolling of a Nonhomogeneous Ball Over a Sphere Without Slipping and Twisting
Abstract
Consider the problem of rolling a dynamically asymmetric balanced ball (the Chaplygin ball) over a sphere. Suppose that the contact point has zero velocity and the projection of the angular velocity to the normal vector of the sphere equals zero. This model of rolling differs from the classical one. It can be realized, in some approximation, if the ball is rubber coated and the sphere is absolutely rough. Recently, J. Koiller and K. Ehlers pointed out the measure and the Hamiltonian structure for this problem. Using this structure we construct an isomorphism between this problem and the problem of the motion of a point on a sphere in some potential field. The integrable cases are found.

Damianou P. A., Papageorgiou V. G.
On an Integrable Case of Kozlov–Treshchev Birkhoff Integrable Potentials
Abstract
We establish, using a new approach, the integrability of a particular case in the Kozlov–Treshchev classification of Birkhoff integrable Hamiltonian systems. The technique used is a modification of the so called quadratic Lax pair for $D_n$ Toda lattice combined with a method used by M. Ranada in proving the integrability of the Sklyanin case.

Davison C. M., Dullin H. R.
Geodesic Flow on ThreeDimensional Ellipsoids with Equal SemiAxes
Abstract
Following on from our previous study of the geodesic flow on three dimensional ellipsoid with equal middle semiaxes, here we study the remaining cases: Ellipsoids with two sets of equal semiaxes with $SO(2) \times SO(2)$ symmetry, ellipsoids with equal larger or smaller semiaxes with $SO(2)$ symmetry, and ellipsoids with three semiaxes coinciding with $SO(3)$ symmetry. All of these cases are Liouvilleintegrable, and reduction of the symmetry leads to singular reduced systems on lowerdimensional ellipsoids. The critical values of the energymomentum maps and their singular fibers are completely classified. In the cases with $SO(2)$ symmetry there are corank 1 degenerate critical points; all other critical points are nondegenreate. We show that in the case with $SO(2) \times SO(2)$ symmetry three global action variables exist and the image of the energy surface under the energymomentum map is a convex polyhedron. The case with $SO(3)$ symmetry is noncommutatively integrable, and we show that the fibers over regular points of the energycasimir map are $T^2$ bundles over $S^2$.

Efimova O. Y., Kudryashov N. A.
Power Expansions for the SelfSimilar Solutions of the Modified Sawada–Kotera Equation
Abstract
The fourthorder ordinary differential equation that denes the selfsimilar solutions of the Kaup–Kupershmidt and Sawada–Kotera equations is studied. This equation belongs to the class of fourthorder analogues of the Painlevé equations. All the power and nonpower asymptotic forms and expansions near points $z = 0$, $z = \infty$ and near an arbitrary point $z = z_0$ are found by means of power geometry methods. The exponential additions to the solutions of the studied equation are also determined.

Chaplygin S. A.
One Case of Vortex Motion in Fluid
Abstract
This text presents an English translation of the significant paper [6] on vortex dynamics
published by the outstanding Russian scientist S.A.Chaplygin, which seems to have almost
escaped the attention of later investigators in this field. Although it was published more than a
century ago, in our opinion it is still interesting and valuable.
