arXiv research

Dancing polygons and rolling balls linked via a special geometric distribution.

A well-known and interesting family of sub-Riemannian space are the systems involving two balls rolling against each other without slipping or twisting. In this note, we show how the sub-Riemannian geodesics of these space, when the two balls are embedded in $\mathbb{R}^3 \times \mathbb{R}^3$, are horizontal curves on …

The paper characterizes surfaces where the speed of a ball is constant.

Understanding the exceptional Lie groups as the symmetry groups of simpler objects is a long-standing program in mathematics. Here, we explore one famous realization of the smallest exceptional Lie group, G2. Its Lie algebra acts locally as the symmetries of a ball rolling on a larger ball, but only when the ratio of r…

We study the rolling of the Chaplygin ball in $\mathbb R^n$ over a fixed $(n-1)$--dimensional sphere without slipping and without slipping and twisting. The problems can be naturally considered within a framework of appropriate modifications of the L+R and LR systems -- well known systems on Lie groups groups with an i…

We relate a Chaplygin type system to a Cartan decomposition of a real semi-simple Lie group. The resulting system is described in terms of the structure theory associated to the Cartan decomposition. It is shown to possess a preserved measure and when internal symmetries are present these are factored out via a process…

We study a time reparametrisation of the Newton type equations on Riemannian manifolds slightly modifying the Chaplygin multiplier method, allowing us to consider the Chaplygin method and the Maupertuis principle within a unified framework. As an example, the reduced nonholonomic problem of rolling without slipping and…

We present the classical Wagner construction from 1935 of the curvature tensor for completely nonholonomic manifolds in both invariant and coordinate way. The starting point is the Shouten curvature tensor for nonholonomic connection introduced by Vranceanu and Shouten. We illustrate the construction on two mechanical …

In this semi-expository paper we disclose hidden symmetries of a classical nonholonomic kinematic model and try to explain geometric meaning of basic invariants of vector distributions.

The Rolling Ball Theorem asserts that given a convex body K in Euclidean space and having a smooth surface bd(K) with all principal curvatures not exceeding c>0 at all boundary points, K necessarily has the property that to each boundary point there exists a ball B_r of radius r=1/c, fully contained in K and touching b…

Innovative ball bearing converts rotary to reciprocating motion.

In this paper, we consider the following general evolution equation $u_t=Δ_fu+au\log^αu+bu$ on smooth metric measure spaces $(M^n, g, e^{-f}dv)$. We give a local gradient estimate of Souplet-Zhang type for positive smooth solution of this equation provided that the Bakry-Émery curvature bounded from below. When $f$…

We prove a Li-Yau gradient estimate for positive solutions to the heat equation, with Neumann boundary conditions, on a compact Riemannian submanifold with boundary ${\bf M}^n\subseteq {\bf N}^n$, satisfying the integral Ricci curvature assumption: \begin{equation} D^2 \sup_{x\in {\bf N}} \left( \oint_{B(x,D)} |Ric^-|^…

Explains rolling of symmetric spaces on flat spaces.

Study on rolling Stiefel manifolds with specific metrics.

A new model for curves on manifolds using rolling operations.

The Blaschke rolling disk theorem is extended to non-convex domains.

Paper extends Noether's Theorem to nonholonomic systems, proving conserved momentum.

Local equivalence found between maximally symmetric rolling and flat Cartan distributions.

Generalized Blaschke rolling theorem for curved spaces.

Rolling two hyperboloid surfaces is described using a Monge normal form.

Shapes can roll downhill following any curve, but often return to initial orientation after crossing multiple copies.

In the present work we define the rolling of one pseudo-Riemannian manifold over another without slipping and twisting. We compare the definition of the rolling without slipping and twisting of two manifolds isometrically embedded into a pseudo-Euclidean space with the rolling defined only by the intrinsic data, namely…

Rollings of reductive homogeneous spaces are studied using intrinsic curves.

New method for rolling bodies on inclined planes, with applications to rescue operations.

Let G be a Lie group. On the trivial principal G-bundle over the Lie algebra of G there is a natural connection whose curvature is the Lie bracket. The exponential map is given by parallel transport of this connection. If G is the diffeomorphism group of a manifold, the curvature of the natural connection is the Lie br…

In the present paper, we study the infinitesimal symmetries of the model of two Riemannian manifolds $(M,g)$ and $(\hat M,\hat g)$ rolling without twisting or slipping. We show that, under certain genericity hypotheses, the natural bundle projection from the state space $Q$ of the rolling model onto $M$ is a principal …

In this article, we consider the rolling (or development) of two Riemannian connected manifolds $(M,g)$ and $(\hat{M},\hat{g})$ of dimensions $2$ and $3$ respectively, with the constraints of no-spinning and no-slipping. The present work is a continuation of \cite{MortadaKokkonenChitour}, which modelled the general set…

Starting from the observation that a flying saucer is a nonholonomic mechanical system whose 5-dimensional configuration space is a contact manifold, we show how to enrich this space with a number of geometric structures by imposing further nonlinear restrictions on the saucer's velocity. These restrictions define cert…

Study rolling control of Lorentzian manifolds on flat space.

We give a complete answer to the question of when two curves in two different Riemannian manifolds can be seen as trajectories of rolling one manifold on the other without twisting or slipping. We show that up to technical hypotheses, a rolling along these curves exists if and only if the geodesic curvatures of each cu…

In this paper, we consider two cases of rolling of one smooth connected complete Riemannian manifold $(M,g)$ onto another one $(\hM,\hg)$ of equal dimension $n\geq 2$. The rolling problem $(NS)$ corresponds to the situation where there is no relative spin (or twist) of one manifold with respect to the other one. As for…

This paper studies hamiltonization of nonholonomic systems using geometric tools. By making use of symmetries and suitable first integrals of the system, we explicitly define a global 2-form for which the gauge transformed nonholonomic bracket gives rise to a new bracket on the reduced space codifying the nonholonomic …

Given any smooth plane curve α(s)representing a mirror that reflects light the usual way and any radiant light source at a point in the plane, the reflected light will produce a caustic envelope. For such an envelope, we show that there is an associated curve \b{eta}(s) and a family of circles C(s) that roll on \b{eta}…

We study a rolling model from the perspective of probability. More precisely, we consider a Riemannian manifold rolling against Euclidean space, where the rolling is coupled with random slipping and twisting. The system is modelled by a stochastic differential equation of Stratonovich-type driven by semimartingales, on…

The study proves properties of optimizers for sets maximizing perimeter under fixed volume constraints.

We present an intrinsic formulation of the kinematic problem of two $n-$dimensional manifolds rolling one on another without twisting or slipping. We determine the configuration space of the system, which is an $\frac{n(n+3)}2-$dimensional manifold. The conditions of no-twisting and no-slipping are decoded by means of …

ERDM integrates rolling forecasts with diffusion models for complex dynamics.

In the present paper we give a historical account -ranging from classical to modern results- of the problem of rolling two Riemannian manifolds one on the other, with the restrictions that they cannot instantaneously slip or spin one with respect to the other. On the way we show how this problem has profited from the d…

In this paper we bring to bear some new tools from statistical learning on the analysis of roll call data. We present a new data-driven model for roll call voting that is geometric in nature. We construct the model by adapting the "Partition Decoupling Method," an unsupervised learning technique originally developed fo…

We use the methods of geometric control theory to study extremal trajectories of vertical rolling disk. We focus on the role of symmetries of the underlying geometric structures. We demonstrate the computations in the CAS Maple package DifferentialGeometry.

New kinematic model for a spin-rolling sphere using Darboux frame.

New definition of Bäcklund transformation for surface isometric deformation.

Study maximally symmetric distribution of An-Nurowski surface rolling on a plane.

Rolling systems limit to billiard models with no-slip collisions.

Rolling Diffusion improves video prediction by progressively corrupting frames based on their temporal position.

The paper uses stochastic control to analyze interest rate markets with roll-over risk.

New proof confirms rolling objects can follow any path.