Joel Langer

Poisson geometry of the filament equation

SummaryThe Hasimoto transformation (relating vortex filament flow to the nonlinear Schrödinger equation) is interpreted in the context of Poisson geometry with the aid of a compact formula for its differential. A useful relationship is derived between Killing fields for soliton solutions of the filament flow and the sequence of commuting Hamiltonian flows.

Lagrangian Aspects of the Kirchhoff Elastic Rod

The relation between Eulers planar elastic curves and vortex filaments evolving by the localized induction equation (LIE) of hydrodynamics was discovered by Hasimoto in 1971. Basic facts about (an integrable case of) Kirchhoff elastic rods are described here, which amplify the connection between the variational problem for rods and the soliton equation LIE. In particular, it is shown that the centerline of the Kirchhoff rod is an equilibrium for a linear combination of the first three conserved Hamiltonians in the LIE hierarchy.

Curve straightening and a minimax argument for closed elastic curves

THE RESULTS obtained here have to do with the following problem. Imagine the ends of a straight length of springy wire are joined together smoothly and the wire is held in some configuration described by an immersion y of the circle into the plane or into R3. According to the Bernoulli-Euler theory of elastic rods the bending energy of the wire is proportional to the total squared curvature of y, which we will denote by F (7) = 5, k2 ds. Suppose now the wire is released and it moves so as to decrease its bending energy as efficiently as possible, i.e. following “the negative gradient of F” (so our dynamics are Aristotelian rather than Newtonian, and we are also making the physically unrealistic assumption that the wire can pass through itself freely). How does the wire evolve, and what will happen ultimately (as time goes to infinity)? Of course, one wants to know first that one can actually define such a flow on the space of immersed circles, that it exists for all time, and that one can sensibly speak of a limiting curve yrn for the trajectory through a given initial curve yo. It is shown here that this is indeed the case and that in fact the Palais-Smale condition holds for this flow. It is proved, moreover, that if y0 is a plane curve of rotation index one (e.g. if y0 is embedded) then the flow carries y0 to a circle. Our main result, however, pertains to the non-planar case, where the situation is more complicated. In a space form it is possible to integrate the equations for an elastica, i.e. for a critical point of F, and this enables one to prove, in particular, that there is a countably infinite family of (similarity classes of) closed elastic curves in R3 (see Theorem 0.1). Thus, not all wire loops in R3 will flow to a circle. On the other hand, this leaves open the possibility that ycu is a circle for almost any initial curve yO, and indeed, our concluding Theorem 3.2 states that the circle is the only stable closed elastica in R3. The proof of this theorem itself depends on the dynamical, i.e. gradient flow approach to the study of F (and avoids a detailed analysis of the Hessian of F, which is quite complicated for non-planar elastic curves). The idea is as follows. One considers a discrete group G of rotations of R3 and an associated pair of multiply covered circular elastic curves which are Gequivariantly regularly homotopic, and which are both local minima for the restriction of F to G-symmetric curves (though multiple circles are unstable with respect to general variations). An appeal to the minimax and symmetric criticality principles then enables one to conclude that there exists a non-circular elastica of “saddle type”. Comparision with the classification theorem shows that one can account in this way for all non-circular solutions, hence all are unstable. We remark that a similar critical structure occurs for “free” (length unconstrained) elastic curves in the standard two-sphere: it was shown in [S] (by an entirely different method) that all closed non-geodesic solutions in S2 are unstable and can be regarded as minimax critical points arising from symmetrical regular homotopies between certain multiple coverings of a prime geodesic (though the minimax argument in [SJ is made only heuristically). To the extent that a similar picture holds as well for manifolds of (non-constant) positive curvature one gains a new view of closed geodesics as the limits of almost all trajectories of -VF. The organization of the paper is as follows. Section 0 is a brief review of some basic facts concerning elastic curves in space forms and the classification of closed elastic curves in R3 (details can be found in [S], [6] ). Section 1 is devoted mostly to the proof ofcondition (C)for the curve straightening flow. We have included details and have attempted to keep the discussion as self-contained as possible. In Section 2 we derive a second variation formula

CURVE MOTION INDUCING MODIFIED KORTEWEG-DE VRIES SYSTEMS

Abstract A straightforward vector generalization of the modified Korteweg-deVries equation is shown to be intimately related to the geometry of curves in n-dimensional Euclidean spaces and spheres. This mKdV system, which is coupled in a particularly a simple way, describes the dynamics of the natural curvature vector of a unit speed curve subject to an elementary geometric evolution equation. The underlying structure of these equations is related to generalizations of the nonlinear Schrodinger and localized induction equations in the context of Hermitian symmetric Lie algebras.

Local geometric invariants of integrable evolution equations

The integrable hierarchy of commuting vector fields for the localized induction equation of 3D hydrodynamics, and its associated recursion operator, are used to generate families of integrable evolution equations which preserve local geometric invariants of the evolving curve or swept‐out surface.

The Hasimoto transformation and integrable flows on curves

Abstract A simple formula for the differential of the Hasimoto transformation (relating vortex filament flow to the cubic Schrodinger equation) is presented.

Taimanov’s surface evolution and Bäcklund transformations for curves

Taimanov’s evolution of conformally parametrized surfaces in Euclidean space by the modified Novikov-Veselov equation is interpreted here (in the revolution case) using hyperbolic geometry and Bäcklund transformations for curves.

Schwarz Reflection Geometry II: Local and Global Behavior of the Exponential Map

A local normal form is obtained for geodesics in the space Λ = {Γ} of analytic Jordan curves in the extended complex plane with symmetric space multiplication Γ1 · Γ2 defined by Schwarzian reflection of Γ2 in Γ1. Local geometric features of (Λ, ·) will be seen to reflect primarily the structure of the Witt algebra, while issues of global behavior of the exponential map will be viewed in the context of conformal mapping theory.

Straightening soliton curves

A canonical straightening process is described for soliton curves associated with the localized induction hierarchy. Following computer animated examples, the present topic is placed in the context of a larger theme: the soliton class is a natural setting for representation of diverse topological and geometrical behavior of curves and their motions.

Rational diffeomorphisms of the circle

Abstract We prove an interpolation theorem for rational circle diffeomorphisms: A set of N complex numbers of unit modulus may be mapped to any corresponding set by a ratio of polynomials p ( z ) / q ( z ) which restricts an invertible mapping of the unit circle S 1 ⊂ C onto itself.