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We study a simple model of bicycle motion: a segment of fixed length in multi-dimensional Euclidean space, moving so that the velocity of the rear end is always aligned with the segment. If the front track is prescribed, the trajectory of the rear wheel is uniquely determined via a certain first order differential equation -- the bicycle equation. The same model, in dimension two, describes another mechanical device, the hatchet planimeter. Here is a sampler of our results. We express the linearized flow of the bicycle equation in terms of the geometry of the rear track; in dimension three, for closed front and rear tracks, this is a version of the Berry phase formula. We show that in all dimensions a sufficiently long bicycle also serves as a planimeter: it measures, approximately, the area bivector defined by the closed front track. We prove that the bicycle equation also describes rolling, without slipping and twisting, of hyperbolic space along Euclidean space. We relate the bicycle problem with two completely integrable systems: the AKNS (Ablowitz, Kaup, Newell and Segur) system and the vortex filament equation. We show that bicycle correspondence of space curves (front tracks sharing a common back track) is a special case of a Darboux transformation associated with the AKNS system. We show that the filament hierarchy, encoded as a single generating equation, describes a 3-dimensional bike of imaginary length. We show that a series of examples of ambiguous closed bicycle curves (front tracks admitting self bicycle correspondence), found recently F. Wegner, are buckled rings, or solitons of the planar filament equation. As a case study, we give a detailed analysis of such curves, arising from bicycle correspondence with multiply traversed circles.
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Louis Poinsot has shown in 1854 that the motion of a rigid body, with one of its points fixed, can be described as the rolling without slipping of one cone, the body cone, along another, the space cone, with their common vertex at the fixed point. This description has been further refined by the second author in 1996, relating the geodesic curvatures of the spherical curves formed by intersecting the cones with the unit sphere in Euclidean $mathbb{R}^3$, thus enabling a reconstruction of the motion of the body from knowledge of the space cone together with the (time dependent) magnitude of the angular velocity vector. In this article we show that a similar description exists for a time dependent family of unimodular $ 2 times 2 $ matrices in terms of rolling cones in 3-dimensional Minkowski space $mathbb{R}^{2,1}$ and the associated pseudo spherical curves, in either the hyperbolic plane $H^2$ or its Lorentzian analog $H^{1,1}$. In particular, this yields an apparently new geometric interpretation of Schrodingers (or Hills) equation $ ddot x + q(t) x =0 $ in terms of rolling without slipping of curves in the hyperbolic plane.
Based on the classical Plucker correspondence, we present algebraic and geometric properties of discrete integrable line complexes in $CP^3$. Algebraically, these are encoded in a discrete integrable system which appears in various guises in the theory of continuous and discrete integrable systems. Geometrically, the existence of these integrable line complexes is shown to be guaranteed by Desargues classical theorem of projective geometry. A remarkable characterisation in terms of correlations of $CP^3$ is also recorded.
A differential 1-form $alpha$ on a manifold of odd dimension $2n+1$, which satisfies the contact condition $alpha wedge (dalpha)^n eq 0$ almost everywhere, but which vanishes at a point $O$, i.e. $alpha (O) = 0$, is called a textit{singular contact form} at $O$. The aim of this paper is to study local normal forms (formal, analytic and smooth) of such singular contact forms. Our study leads naturally to the study of normal forms of singular primitive 1-forms of a symplectic form $omega$ in dimension $2n$, i.e. differential 1-forms $gamma$ which vanish at a point and such that $dgamma = omega$, and their corresponding conformal vector fields. Our results are an extension and improvement of previous results obtained by other authors, in particular Lychagin cite{Lychagin-1stOrder1975}, Webster cite{Webster-1stOrder1987} and Zhitomirskii cite{Zhito-1Form1986,Zhito-1Form1992}. We make use of both the classical normalization techniques and the toric approach to the normalization problem for dynamical systems cite{Zung_Birkhoff2005, Zung_Integrable2016, Zung_AA2018}.
In this paper, we will provide a review of the geometric construction, proposed by Witten, of the SU(n) quantum representations of the mapping class groups which are part of the Reshetikhin-Turaev TQFT for the quantum group U_q(sl(n, C)). In particular, we recall the differential geometric construction of Hitchins projectively flat connection in the bundle over Teichmuller space obtained by push-forward of the determinant line bundle over the moduli space of rank n, fixed determinant, semi-stable bundles fibering over Teichmuller space. We recall the relation between the Hitchin connection and Toeplitz operators which was first used by the first named author to prove the asymptotic faithfulness of the SU(n) quantum representations of the mapping class groups. We further review the construction of the formal Hitchin connection, and we discuss its relation to the full asymptotic expansion of the curve operators of Topological Quantum Field Theory. We then go on to identifying the first terms in the formal parallel transport of the Hitchin connection explicitly. This allows us to identify the first terms in the resulting star product on functions on the moduli space. This is seen to agree with the first term in the star product on holonomy functions on these moduli spaces defined by Andersen, Mattes and Reshetikhin.
We consider discrete analogues of two well-known open problems regarding invariant measures for dispersive PDE, namely, the invariance of the Gibbs measure for the continuum (classical) Heisenberg model and the invariance of white noise under focusing cubic NLS. These continuum models are completely integrable and connected by the Hasimoto transform; correspondingly, we focus our attention on discretizations that are also completely integrable and also connected by a discrete Hasimoto transform. We consider these models on the infinite lattice $mathbb Z$. Concretely, for a completely integrable variant of the classical Heisenberg spin chain model (introduced independently by Haldane, Ishimori, and Sklyanin) we prove the existence and uniqueness of solutions for initial data following a Gibbs law (which we show is unique) and show that the Gibbs measure is preserved under these dynamics. In the setting of the focusing Ablowitz--Ladik system, we prove invariance of a measure that we will show is the appropriate discrete analogue of white noise. We also include a thorough discussion of the Poisson geometry associated to the discrete Hasimoto transform introduced by Ishimori that connects the two models studied in this article.
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