Jair Koiller
Current Organizations / Institutes
The organizations/institutes from the scholar's latest article
Previous Organizations / Institutes
The organizations / institutes from the scholar's other articles in the past ten years
Research Subjects
Jair Koiller has a total of 24 co-authors. Here are the top co-authors by number of shared publications:
- Dan Reznik
Data Science Consulting
8 shared publications - Ronaldo Garcia
Instituto de Matemática e Estatística, Universidade Federal de Goiás, Goiânia, Brazil
8 shared publications - Kurt Ehlers
Mathematics Department, Truckee Meadows Community College
5 shared publications - Teresinha J. Stuchi
Universidade Federal do Rio de Janeiro
4 shared publications - Alejandro Cabrera
Universidade Federal do Rio de Janeiro
3 shared publications - César Castilho
Universidade Federal de Pernambuco
3 shared publications
Latest Publications
- 20 December 2025
- Published by Pleiades Publishing Ltd in Regular and Chaotic Dynamics
- Vol. 31 (1) , 1-27
Comments on a Paper about Rubber Rolling by A. V. Borisov, I. S. Mamaev and I. A. Bizyaev (with an Appendix by Luis C. García-Naranjo)
Rubber rolling (meaning no-slip and no-twist constraints) of a convex body on the plane under the influence of gravity is a \(SE(2)\) Chaplygin system that reduces to the cotangent bundle of the unit sphere of Poisson vectors. I comment here upon an observation by A. V. Borisov and I. S. Mamaev [1, 2008], also found in A. V. Borisov, I. S. Mamaev and I. A. Bizyaev [2, 2013] that surfaces of revolution are special: the additional integral of motion is elementary, while for marble rolling it requires special functions. I use the term “Nose function” to refer to their expression \(N(\theta)=\big{(}I_{1}\cos^{2}\theta+I_{3}\sin^{2}\theta+mz_{C}^{2}(\theta)\big{)}^{1/2}\) where \(\theta\) is the nutation and \(z_{C}(\theta)\) is the center of mass height. \(N(\theta)\) appears somewhat miraculously in the process of the almost symplectic reduction. I work in a space frame using the Euler angles \(\phi \text{(yaw)}, \psi \text{ roll and } \theta\). The reduction to 1 DoF is done in two stages: first, reduction by the group \(SE(2)=\{(x,y,\phi)\}\) to \(T^{*}S^{2}\) with almost symplectic 2-form \(\Omega_{NH}=dp_{\theta}\wedge d\theta+dp_{\psi}\wedge d\psi+J\cdot K\). The semibasic term is \(J\cdot K=-p_{\psi}(d\log\big{(}N(\theta)\big{)}\wedge d\psi\). It follows that \(\Omega_{NH}\) is conformally symplectic in the sense that \(d\left(\frac{1}{N}\Omega_{NH}\right)=0.\) The conserved quantity due to the \(S^{1}\) symmetry about the body axis is \(\ell=N(\theta)\sin^{2}\theta\dot{\psi}\), yielding the desired reduction to \((\theta,p_{\theta})\). Further simplification results by taking the new time \(dt=\sqrt{B(\theta)}d\tau, \text{ with } B=I_{1}+m|CP|^{2}\) where \(P=(x,y)\) is the point of contact. One gets finally \(H=\frac{1}{2}\tilde{p}^{2}_{\theta}+V(\theta),V(\theta)=\ell^{2}/2\sin^{2}\theta+mgz_{C}(\theta) \text{ with } \tilde{p}_{\theta}=p_{\theta}/\sqrt{B}\) and usual symplectic form \(d\tilde{p}_{\theta}\wedge d\theta\). The moments of inertia \(I_{1},I_{3}\) reappear in the reconstruction. As an example, very basic observations are presented for the torus. A detailed study was just finished by A. Kilin and E. Pivovarova in [3].- 20 December 2025
- Published by Pleiades Publishing Ltd in Regular and Chaotic Dynamics
- Vol. 31 (1) , 1-27
Comments on a Paper about Rubber Rolling by A. V. Borisov, I. S. Mamaev and I. A. Bizyaev (with an Appendix by Luis C. García-Naranjo)
Rubber rolling (meaning no-slip and no-twist constraints) of a convex body on the plane under the influence of gravity is a \(SE(2)\) Chaplygin system that reduces to the cotangent bundle of the unit sphere of Poisson vectors. I comment here upon an observation by A. V. Borisov and I. S. Mamaev [1, 2008], also found in A. V. Borisov, I. S. Mamaev and I. A. Bizyaev [2, 2013] that surfaces of revolution are special: the additional integral of motion is elementary, while for marble rolling it requires special functions. I use the term “Nose function” to refer to their expression \(N(\theta)=\big{(}I_{1}\cos^{2}\theta+I_{3}\sin^{2}\theta+mz_{C}^{2}(\theta)\big{)}^{1/2}\) where \(\theta\) is the nutation and \(z_{C}(\theta)\) is the center of mass height. \(N(\theta)\) appears somewhat miraculously in the process of the almost symplectic reduction. I work in a space frame using the Euler angles \(\phi \text{(yaw)}, \psi \text{ roll and } \theta\). The reduction to 1 DoF is done in two stages: first, reduction by the group \(SE(2)=\{(x,y,\phi)\}\) to \(T^{*}S^{2}\) with almost symplectic 2-form \(\Omega_{NH}=dp_{\theta}\wedge d\theta+dp_{\psi}\wedge d\psi+J\cdot K\). The semibasic term is \(J\cdot K=-p_{\psi}(d\log\big{(}N(\theta)\big{)}\wedge d\psi\). It follows that \(\Omega_{NH}\) is conformally symplectic in the sense that \(d\left(\frac{1}{N}\Omega_{NH}\right)=0.\) The conserved quantity due to the \(S^{1}\) symmetry about the body axis is \(\ell=N(\theta)\sin^{2}\theta\dot{\psi}\), yielding the desired reduction to \((\theta,p_{\theta})\). Further simplification results by taking the new time \(dt=\sqrt{B(\theta)}d\tau, \text{ with } B=I_{1}+m|CP|^{2}\) where \(P=(x,y)\) is the point of contact. One gets finally \(H=\frac{1}{2}\tilde{p}^{2}_{\theta}+V(\theta),V(\theta)=\ell^{2}/2\sin^{2}\theta+mgz_{C}(\theta) \text{ with } \tilde{p}_{\theta}=p_{\theta}/\sqrt{B}\) and usual symplectic form \(d\tilde{p}_{\theta}\wedge d\theta\). The moments of inertia \(I_{1},I_{3}\) reappear in the reconstruction. As an example, very basic observations are presented for the torus. A detailed study was just finished by A. Kilin and E. Pivovarova in [3].- Open Access
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).
Open AccessConformally 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).
- 1 August 2025
- Published by American Mathematical Society (AMS) in Notices of the American Mathematical Society
- Vol. 72 (07)
Apropos the Long-Lost Letter from Levinson to Smale, Now Found!
- 1 August 2025
- Published by American Mathematical Society (AMS) in Notices of the American Mathematical Society
- Vol. 72 (07)
Apropos the Long-Lost Letter from Levinson to Smale, Now Found!
- Open Access
- 1 January 2025
- Published by National Library of Serbia in Theoretical and Applied Mechanics
- Vol. 52 (2) , 195-216
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 grad)u = -grad p , \operatorname{div}(bu) = 0 , with u \parallel \partial D . It is well posed as a PDE, but when b \neq \operatorname{const} , justifying point vortex models requires the analyst?s attention. We focus on Geometric Mechanics aspects, glossing over hard analysis issues. The motivating example is a ?rip current? produced by vortex pairs near a beach shore. For a beach with uniform slope, 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 -\operatorname{div}(grad \psi / b) with Dirichlet boundary conditions. As in elasticity, the lake equations give rise to pseudo-analytical functions and quasi-conformal mappings. Uniformly elliptic equations on close Riemann surfaces could be called ?planet equations?.
- 1 January 2025
- Published by National Library of Serbia in Theoretical and Applied Mechanics
- Vol. 52 (2) , 195-216
Open AccessVortices 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 grad)u = -grad p , \operatorname{div}(bu) = 0 , with u \parallel \partial D . It is well posed as a PDE, but when b \neq \operatorname{const} , justifying point vortex models requires the analyst?s attention. We focus on Geometric Mechanics aspects, glossing over hard analysis issues. The motivating example is a ?rip current? produced by vortex pairs near a beach shore. For a beach with uniform slope, 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 -\operatorname{div}(grad \psi / b) with Dirichlet boundary conditions. As in elasticity, the lake equations give rise to pseudo-analytical functions and quasi-conformal mappings. Uniformly elliptic equations on close Riemann surfaces could be called ?planet equations?.
- 1 February 2024
- Published by Pleiades Publishing Ltd in Regular and Chaotic Dynamics
- Vol. 29 (2) , 241-303
On the Interplay Between Vortices and Harmonic Flows: Hodge Decomposition of Euler’s Equations in 2d
Let \(\Sigma\) be a compact manifold without boundary whose first homology is nontrivial. The Hodge decomposition of the incompressible Euler 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 nonflat metric that shows typical features of nonintegrable 2 degrees of freedom Hamiltonians. In contradistinction, for flat tori the harmonic part is constant. Next, we turn to hyperbolic surfaces (\(g\geqslant 2\)) 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 hydrodynamic Green function is clarified. A number of questions are suggested.- 1 February 2024
- Published by Pleiades Publishing Ltd in Regular and Chaotic Dynamics
- Vol. 29 (2) , 241-303
On the Interplay Between Vortices and Harmonic Flows: Hodge Decomposition of Euler’s Equations in 2d
Let \(\Sigma\) be a compact manifold without boundary whose first homology is nontrivial. The Hodge decomposition of the incompressible Euler 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 nonflat metric that shows typical features of nonintegrable 2 degrees of freedom Hamiltonians. In contradistinction, for flat tori the harmonic part is constant. Next, we turn to hyperbolic surfaces (\(g\geqslant 2\)) 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 hydrodynamic Green function is clarified. A number of questions are suggested.- Open Access
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.Open AccessOn 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. - 1 September 2022
- Published by Pleiades Publishing Ltd in Regular and Chaotic Dynamics
- Vol. 27 (5) , 493-524
On the Linear Stability of a Vortex Pair Equilibrium on a Riemann Surface of Genus Zero
We present a simple procedure to perform the linear stability analysis of a vortex pair equilibrium on a genus zero surface with an arbitrary metric. It consists of transferring the calculations to the round sphere in \(\mathbb{R}^{3}\), with a conformal factor, and exploring the Möbius invariance of the conformal structure, so that the equilibria, seen on the representing sphere, appear in the north/south poles. Three example problems are analyzed: \(i)\) For a surface of revolution of genus zero, a vortex pair located at the poles is nonlinearly stable due to integrability. We compute the two frequencies of the linearization. One is for the reduced system, the other is related to the reconstruction. Exceptionally, one of them can vanish. The calculation requires only the local profile at the poles and one piece of global information (given by a quadrature). \(ii)\) A vortex pair on a double faced elliptical region, limiting case of the triaxial ellipsoid when the smaller axis goes to zero. We compute the frequencies of the pair placed at the centers of the faces. \(iii)\) The stability, to a restricted set of perturbations, of a vortex equilateral triangle located in the equatorial plane of a spheroid, with polar vortices added so that the total vorticity vanishes.- 1 September 2022
- Published by Pleiades Publishing Ltd in Regular and Chaotic Dynamics
- Vol. 27 (5) , 493-524
On the Linear Stability of a Vortex Pair Equilibrium on a Riemann Surface of Genus Zero
We present a simple procedure to perform the linear stability analysis of a vortex pair equilibrium on a genus zero surface with an arbitrary metric. It consists of transferring the calculations to the round sphere in \(\mathbb{R}^{3}\), with a conformal factor, and exploring the Möbius invariance of the conformal structure, so that the equilibria, seen on the representing sphere, appear in the north/south poles. Three example problems are analyzed: \(i)\) For a surface of revolution of genus zero, a vortex pair located at the poles is nonlinearly stable due to integrability. We compute the two frequencies of the linearization. One is for the reduced system, the other is related to the reconstruction. Exceptionally, one of them can vanish. The calculation requires only the local profile at the poles and one piece of global information (given by a quadrature). \(ii)\) A vortex pair on a double faced elliptical region, limiting case of the triaxial ellipsoid when the smaller axis goes to zero. We compute the frequencies of the pair placed at the centers of the faces. \(iii)\) The stability, to a restricted set of perturbations, of a vortex equilateral triangle located in the equatorial plane of a spheroid, with polar vortices added so that the total vorticity vanishes.- 10 August 2022
- Published by Springer Nature in Journal of Dynamical and Control Systems
- Vol. 29 (3) , 757-767
Estimating Elliptic Billiard Invariants with Spatial Integrals
- 10 August 2022
- Published by Springer Nature in Journal of Dynamical and Control Systems
- Vol. 29 (3) , 757-767
Estimating Elliptic Billiard Invariants with Spatial Integrals
- 21 June 2022
- Published by Elsevier in Journal of Symbolic Computation
- Vol. 114, 336-358
Loci of 3-periodics in an Elliptic Billiard: Why so many ellipses?
- 21 June 2022
- Published by Elsevier in Journal of Symbolic Computation
- Vol. 114, 336-358
Loci of 3-periodics in an Elliptic Billiard: Why so many ellipses?
- Open Access
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
Open AccessLoci 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