arXiv subjects

Jair Koiller

Publications and source records attributed to Jair Koiller.

16 recordsLinked to original sources

arXiv2025

Conformally symplectic Chaplygin reduction in rubber rolling of surfaces of revolution over the plane

Rubber rolling (no-slip and no-twist) of a convex body on the plane under the influence of gravity is a SE(2) Chaplygin system, that reduces to the sphere of Poisson vectors. I comment upon an observation by A.V Borisov and I.S. Mamaev (Regular and Chaotic Dynamics, 13(5):443-490, 2008) for the case of surfaces of revolution [also in A. V. Borisov, I. S. Mamaev and I. A. Bizyaev (Regular and Chaotic Dynamics, 18(3):277-328, 2013)]. They show that this case is quite special: the additional integral of motion is elementary, while for marble rolling it is not elementary. I connect this finding with recent work about Chaplygin reduced systems that are conformally symplectic (Luis Garcia Naranjo and Juan C. Marrero. The geometry of nonholonomic Chaplygin systems revisited. Nonlinearity, 33(3):1297, 2020).

math-ph↗

arXiv2025

Vortices for lake equations (review with questions and speculations)

The `lake equation' on a planar domain D with bathymetry b(x,y) is given by $ \partial_t u + (u \cdot {\rm grad}) u= -{\rm grad}\, p \,, \,\,{\rm div} (b u) = 0 \,,\, \text{with}\,\, u \parallel \partial D.$ % \, \,\, \,\,\, \text{),}$$ We focus on Geometric Mechanics aspects, glossing over hard analysis issues. % related to the desingularization. Motivating example is a `rip current' produced by vortex pairs near a beach shore. For uniform slope beach there is a perfect analogy with \ Thomson's vortex rings. The stream function produced by a vortex is defined as the Green function of the operator $- {\rm div} ( {\rm grad} ψ/b)$ with Dirichlet boundary conditions. As in elasticity, the lake equations give rise to pseudoanalytical functions and quasiconformal mappings. Uniformly elliptic equations on closed Riemann surfaces could be called `planet equations'.

math-ph↗

arXiv2023

On the interplay between vortices and harmonic flows: Hodge decomposition of Euler's equations in 2d

Let $Σ$ be a compact manifold without boundary whose first homology is nontrivial. Hodge decomposition of the incompressible Euler's equation in terms of 1-forms yields a coupled PDE-ODE system. The $L^2$-orthogonal components are a `pure' vorticity flow and a potential flow (harmonic, with the dimension of the homology). In this paper we focus on $N$ point vortices on a compact Riemann surface without boundary of genus $g$, with a metric chosen in the conformal class. The phase space has finite dimension $2N+ 2g$. We compute a surface of section for the motion of a single vortex ($N=1$) on a torus ($g=1$) with a non-flat metric, that shows typical features of non-integrable 2-dof Hamiltonians. In contradistinction, for flat tori the harmonic part is constant. Next, we turn to hyperbolic surfaces ($ g \geq 2$), having constant curvature -1, with discrete symmetries. Fixed points of involutions yield vortex crystals in the Poincaré disk. Finally we consider multiply connected planar domains. The image method due to Green and Thomson is viewed in the Schottky double. The Kirchhoff-Routh hamiltonian given in C.C. Lin's celebrated theorem is recovered by Marsden-Weinstein reduction from $2N+2g$ to $2N$. The relation between the electrostatic Green function and the hydrodynamical Green function is clarified. A number of questions are suggested.

math-ph↗

arXiv2022

Loci of 3-periodics in an Elliptic Billiard: why so many ellipses?

A triangle center such as the incenter, barycenter, etc., is specified by a function thrice- and cyclically applied on sidelengths and/or angles. Consider the 1d family of 3-periodics in the elliptic billiard, and the loci of its triangle centers. Some will sweep ellipses, and others higher-degree algebraic curves. We propose two rigorous methods to prove if the locus of a given center is an ellipse: one based on computer algebra, and another based on an algebro-geometric method. We also prove that if the triangle center function is rational on sidelengths, the locus is algebraic

math.DS↗

arXiv2021

New Properties of Triangular Orbits in Elliptic Billiards

New invariants in the one-dimensional family of 3-periodic orbits in the elliptic billiard were introduced by the authors in "Can the Elliptic Billiard Still Surprise Us?" (2020), Math. Intelligencer, 42(1): 6--17, some of which were generalized to $N>3$. Invariants mentioned there included ratios of radii and/or areas, sum of angle cosines, and a special stationary circle. Here we present some of the proofs omitted there as well as a few new related facts.

math.DS↗

arXiv2021

Average Elliptic Billiard Invariants with Spatial Integrals

We compare invariants of N-periodic trajectories in the elliptic billiard, classic and new, to their aperiodic counterparts via a spatial integrals evaluated over the boundary of the elliptic billiard. The integrand is weighed by a universal measure equal to the density of rays hitting a given boundary point. We find that aperiodic averages are smooth and monotonic on caustic eccentricity, and perfectly match N-periodic average invariants at the discrete caustic parameters which admit a given N-periodic family.

math.DS↗

arXiv2020

Eighty New Invariants of N-Periodics in the Elliptic Billiard

We introduce several-dozen experimentally-found invariants of Poncelet N-periodics in the confocal ellipse pair (Elliptic Billiard). Recall this family is fully defined by two integrals of motion (linear and angular momentum), so any "new" invariants are dependent upon them. Nevertheless, proving them may require sophisticated methods. We reference some two-dozen proofs already contributed. We hope this article will motivate contributions for those still lacking proof.

math.DS↗

arXiv2020

The Ballet of Triangle Centers on the Elliptic Billiard

The dynamic geometry of the family of 3-periodics in the Elliptic Billiard is mystifying. Besides conserving perimeter and the ratio of inradius-to-circumradius, it has a stationary point. Furthermore, its triangle centers sweep out mesmerizing loci including ellipses, quartics, circles, and a slew of other more complex curves. Here we explore a bevy of new phenomena relating to (i) the shape of 3-periodics and (ii) the kinematics of certain Triangle Centers constrained to the Billiard boundary, specifically the non-monotonic motion some can display with respect to 3-periodics. Hypnotizing is the joint motion of two such non-monotonic Centers, whose many stops-and-gos are akin to a Ballet.

math.DS↗

arXiv2020

Can the Elliptic Billiard Still Surprise Us?

Can any secrets still be shed by that much studied, uniquely integrable, Elliptic Billiard? Starting by examining the family of 3-periodic trajectories and the loci of their Triangular Centers, one obtains a beautiful and variegated gallery of curves: ellipses, quartics, sextics, circles, and even a stationary point. Secondly, one notices this family conserves an intriguing ratio: Inradius-to-Circumradius. In turn this implies three conservation corollaries: (i) the sum of bounce angle cosines, (ii) the product of excentral cosines, and (iii) the ratio of excentral-to-orbit areas. Monge's Orthoptic Circle's close relation to 4-periodic Billiard trajectories is well-known. Its geometry provided clues with which to generalize 3-periodic invariants to trajectories of an arbitrary number of edges. This was quite unexpected. Indeed, the Elliptic Billiard did surprise us!

math.DS↗

arXiv2017

About simple variational splines from the Hamiltonian viewpoint

In this paper, we study simple splines on a Riemannian manifold $Q$ from the point of view of the Pontryagin maximum principle (PMP) in optimal control theory. The control problem consists in finding smooth curves matching two given tangent vectors with the control being the curve's acceleration, while minimizing a given cost functional. We focus on cubic splines (quadratic cost function) and on time-minimal splines (constant cost function) under bounded acceleration. We present a general strategy to solve for the optimal hamiltonian within the PMP framework based on splitting the variables by means of a linear connection. We write down the corresponding hamiltonian equations in intrinsic form and study the corresponding hamiltonian dynamics in the case $Q$ is the $2$-sphere. We also elaborate on possible applications, including landmark cometrics in computational anatomy.

math.SG↗

arXiv2011

Non-Holonomic Connections Following Élie Cartan

In this note we revisit E. Cartan's address at the 1928 International Congress of Mathematicians at Bologna, Italy. The distributions considered here will be of the same class as those considered by Cartan, a special type which we call strongly non-holonomic. We set up the groundwork for using Cartan's method of equivalence (a powerful tool for obtaining invariants associated to geometrical objects), to more general non-holonomic distributions.

gr-qc↗

arXiv2009

Synechococcus as a "singing" bacterium: biology inspired by micro-engineered acoustic streaming devices

Certain cyanobacteria, such as open ocean strains of Synechococcus, are able to swim at speeds up to 25 diameters per second, without flagella or visible changes in shape. The means by which Synechococcus generates thrust for self-propulsion is unknown. The only mechanism that has not been ruled out employs tangential waves of surface deformations. In Ehlers et al, the average swimming velocity for this mechanism was estimated using the methods inaugurated by Taylor and Lighthill in the 1950's and revisited in differential geometric language by Shapere and Wilczek in 1989. In this article we propose an entirely different physical principle self propulsion based on acoustic streaming (AS). Micro-pumps in silicon chips, based on AS, have been constructed by engineers since the 1990's, but to the best of our knowledge acoustic streaming as a means of microorganisms locomotion has not been proposed before. Our hypothesis is supported by two recent discoveries: (1) In Samuel, et al, deep-freeze electron microscopy of the motile strain WH8113 revealed a crystalline outer layer (CS) covered with a forest of "spicules" (Sp) extending from the inner membrane through the CS, projecting 150 nm into the surrounding fluid. (2) In Pelling et al, atomic force microscopy (AFM) was used to find that the cell wall of yeast cells periodically oscillates on nano-scale amplitudes at frequencies of 0.8 to 1.6 kHz, and that the oscillations are generated metabolically. We propose that the spicules, in contact with the cell's power systems, could perform high frequency motions generating acoustic streaming (AS) in the surrounding fluid. We compare two models for self-propulsion employing acoustic streaming: the quartz wind effect (QW) and boundary induced streaming generated by surface acoustic waves (SAW).

physics.bio-ph↗

arXiv2008

Vortices on closed surfaces

We consider $N$ point vortices $s_j$ of strengths $κ_j$ moving on a closed (compact, boundaryless, orientable) surface $S$ with riemannian metric $g$. As far as we know, only the sphere or surfaces of revolution, the latter qualitatively, have been treated in the available literature. The aim of this note is to present an intrinsic geometric formulation for the general case. We give a simple proof of Kimura's conjecture that a dipole describes geodesic motion. Searching for integrable vortex pairs systems on Liouville surfaces is in order. The vortex pair system on a triaxial ellipsoid extends Jacobi's geodesics. Is it Arnold-Liouville integrable? Not in our wildest dreams is another possibility: that quantizing a vortex system could relate with a million dollars worth question, but we took courage - nerve is more like it - to also present it.

math.SG↗

arXiv2006

Rubber rolling over a sphere

``Rubber'' coated rolling bodies satisfy a no-twist in addition to the no slip satisfied by ``marble'' coated bodies. Rubber rolling has an interesting differential geometric appeal because the geodesic curvatures of the curves on the surfaces at the corresponding points are equal. The associated distribution in the 5 dimensional configuration space has 2-3-5 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 exceptionalgroup G_2. The 2-3-5 nonholonomic geometries are classified in a companion paper via Cartan's equivalence method. Rubber rolling of a convex body over a sphere defines a generalized Chaplygin system with SO(3) symmetry group, total space Q = SO(3) X S^2 that can be reduced to an almost Hamiltonian system in T^*S^2 with a non-closed 2-form ω_{NH}. In this paper we present some basic results on this reduction and as an example we discuss the sphere-sphere problem. In this example the 2-form is conformally symplectic so the reduced system becomes Hamiltonian after a coordinate dependent change of time. In particular there is an invariant measure. Using sphero-conical coordinates we verify the results by Borisov and Mamaev that the system is integrable for a ball over a plane and a rubber ball with twice the radius of a fixed internal ball.

math.SG↗

arXiv2004

Nonholonomic systems via moving frames: Cartan equivalence and Chaplygin Hamiltonization

A nonholonomic system consists of a configuration space Q, a Lagrangian L, and an nonintegrable constraint distribution H, with dynamics governed by Lagrange-d'Alembert's principle. We present two studies both using adapted moving frames. In the first study we apply Cartan's method of equivalence to investigate the geometry underlying a nonholonomic system. As an example we compute the differential invariants for a nonholonomic system on a four-dimensional configuration manifold endowed with a rank two (Engel) distribution. In the second part we study G-Chaplygin systems. These are systems where the constraint distribution is given by a connection on a principal fiber bundle with total space Q and base space S=Q/G, and with a G-equivariant Lagrangian. These systems compress to an almost Hamiltonian system on $T^{*}S$. Under an $s \in S$ dependent time reparameterization a number of compressed systems become Hamiltonian. A necessary condition for Hamiltonization is the existence of an invariant measure on the original system. Assuming an invariant measure we describe the obstruction to Hamiltonization. Chaplygin's "rubber" sphere, a ball with unequal inertia coefficients rolling without slipping or spinning (about the vertical axis) on a plane is Hamiltonizable when compressed to $T^{*}SO(3)$. Finally we discuss reduction of internal symmetries. Chaplygin's "marble" (where spinning is allowed) is not Hamiltonizable when compressed to $T^{*}SO(3)$; we conjecture that it is also not Hamiltonizable when reduced to $T^{*}S^{2}$.

math-ph↗

arXiv2002

Nonholonomic systems with symmetry allowing a conformally symplectic reduction

Non-holonomic mechanical systems can be described by a degenerate almost-Poisson structure (dropping the Jacobi identity) in the constrained space. If enough symmetries transversal to the constraints are present, the system reduces to a nondegenerate almost-Poisson structure on a ``compressed'' space. Here we show, in the simplest non-holonomic systems, that in favorable circumnstances the compressed system is conformally symplectic, although the ``non-compressed'' constrained system never admits a Jacobi structure (in the sense of Marle et al.).

math-ph↗