Tadashi Tokieda

Periodic motions of vortices on surfaces with symmetry

The theory of point vortices in a two-dimensional ideal fluid has a long history, but on surfaces other than the plane no method of finding periodic motions (except relative equilibria) of N vortices is known. We present one such method and find infinite families of periodic motions on surfaces possessing certain symmetries, including spheres, ellipsoids of revolution and cylinders. Our families exhibit bifurcations. N can be made arbitrarily large. Numerical plots of bifurcations are given.

Vortex Dynamics on a Cylinder

Point vortices on a cylinder (periodic strip) are studied geometrically. The Hamiltonian formalism is developed, a nonexistence theorem for relative equilibria is proved, equilibria are classified when all vorticities have the same sign, and several results on relative periodic orbits are established, including as corollaries classical results on vortex streets and leapfrogging.

On relative normal modes

Abstract We generalize the Weinstein-Moser theorem on the existence of nonlinear normal modes near an equilibrium in a Hamiltonian system to a theorem on the existence of relative periodic orbits near a relative equilibrium in a Hamiltonian system with continuous symmetries. In particular, we prove that under appropriate hypotheses there exist relative periodic orbits near relative equilibria even when these relative equilibria are singular points of the corresponding moment map, i.e. when the reduced spaces are singular.

Tides in astronomy and astrophysics

Tides: A Tutorial.- Investigations of Tides From the Antiquity to Laplace.- Ocean Tides.- Precession and Nutation of the Earth.- Tidal Effects of Giant Planets on their Satellites.- Recent Developments in Planet Migration Theory.- Tides in Planetary Systems.- Stellar Tides.- Tides in Colliding Galaxies.

A bug on a raft: recoil locomotion in a viscous fluid

The locomotion of a body through an inviscid incompressible fluid, such that the flow remains irrotational everywhere, is known to depend on inertial forces and on both the shape and the mass distribution of the body. In this paper we consider the influence of fluid viscosity on such inertial modes of locomotion. In particular we consider a free body of variable shape and study the centre-of-mass and centre-of-volume variations caused by a shifting mass distribution. We call this recoil locomotion . Numerical solutions of a finite body indicate that the mechanism is ineffective in Stokes flow but that viscosity can significantly increase the swimming speed above the inviscid value once Reynolds numbers are in the intermediate range 50–300. To study the problem analytically, a model which is an analogue of Taylors swimming sheet is introduced. The model admits analysis at fixed, arbitrarily large Reynolds number for deformations of sufficiently small amplitude. The analysis confirms the significant increase of swimming velocity above the inviscid value at intermediate Reynolds numbers.

Openness of momentum maps and persistence of extremal relative equilibria

We prove that for every proper Hamiltonian action of a Lie group G in finite dimensions the momentum map is locally G-open relative to its image (i.e. images of G-invariant open sets are open). As an application we deduce that in a Hamiltonian system with continuous Hamiltonian symmetries, extremal relative equilibria persist for every perturbation of the value of the momentum map, provided the isotropy subgroup of this value is compact. We also demonstrate how this persistence result applies to an example of ellipsoidal figures of rotating fluid. We also provide an example with plane point vortices which shows how the compactness assumption is related to persistence.

Celt reversals: a prototype of chiral dynamics

(MS received 26 May 2006; accepted 10 January 2007)A physically transparent and mathematically streamlined derivation is presented fora third-order nonlinear dynamical system that describes the curious chiral reversalsof a celt (rattleback). The system is integrable, and its solutions are periodic,showing an infinite succession of spin reversals. Inclusion of linear dissipation allowsany given number of reversals, and a typical celt’s observed behaviour is wellcaptured by tuning the dissipation parameters.

Mechanical Ideas in Geometry

(1998). Mechanical Ideas in Geometry. The American Mathematical Monthly: Vol. 105, No. 8, pp. 697-703.

Deformation of Geometry and Bifurcations of Vortex Rings

We construct a smooth family of Hamiltonian systems, together with a family of group symmetries and momentum maps, for the dynamics of point vortices on surfaces parametrized by the curvature of the surface. Equivariant bifurcations in this family are characterized, whence the stability of the Thomson heptagon is deduced without recourse to the Birkhoff normal form, which has hitherto been a necessary tool.

Rattleback: A model of how geometric singularity induces dynamic chirality

Abstract The rattleback is a boat-shaped top with an asymmetric preference in spin. Its dynamics can be described by nonlinearly coupled pitching, rolling, and spinning modes. The chirality, designed into the body as a skewed mass distribution, manifests itself in the quicker transition of +spin → pitch → −spin than that of −spin → roll → +spin. The curious guiding idea of this work is that we can formulate the dynamics as if a symmetric body were moving in a chiral space. By elucidating the duality of matter and space in the Hamiltonian formalism, we attribute asymmetry to space. The rattleback is shown to live in the space dictated by the Bianchi type VI h − 1 (belonging to class B) algebra; this particular algebra is used here for the first time in a mechanical example. The class B algebra has a singularity that separates the space (Poisson manifold) into asymmetric subspaces, breaking the time-reversal symmetry of nearby orbits.