Jair Koiller

Nonholonomic systems via moving frames: Cartan equivalence and Chaplygin Hamiltonization

A nonholonomic system, for short “NH,” consists of a configuration space Q n, a Lagrangian \( L(q,\dot q,t) \), a nonintegrable constraint distribution \( \mathcal{H} \subset TQ \), with dynamics governed by Lagrange-d’Alembert’s principle. We present here two studies, both using adapted moving frames. In the first we explore the affine connection viewpoint. For natural Lagrangians L = T − V, where we take V = 0 for simplicity, NH-trajectories are geodesics of a (nonmetric) connection ∇N H which mimics Levi-Civita’s. Local geometric invariants are obtained by Cartan’s method of equivalence. As an example, we analyze Engel’s (2–4) distribution. This is the first such study for a distribution that is not strongly nonholonomic. In the second part we study G-Chaplygin systems; for those, the constraints are given by a connection φ: T Q → Lie(G) on a principal bundle G ↪ Q → S = Q/G and the Lagrangian L is G-equivariant. These systems compress to an almost Hamiltonian system (T*S, H φ, ΩN H), ΩN H = Ωcan + (J.K), with d(J.K) ≠ = 0 in general; the momentum map J : T*Q → Lie(G) and the curvature form K : T Q → Lie(G)* are matched via the Legendre transform. Under an s e S dependent time reparametrization, a number of compressed systems become Hamiltonian, i.e., ΩN H is sometimes conformally symplectic. Anecessary condition is the existence of an invariant volume for the original system. Its density produces a candidate for conformal factor. Assuming an invariant volume, we describe the obstruction to Hamiltonization. An example of a Hamiltonizable system is the “rubber” Chaplygin’s sphere, which extends Veselova’s system in T*S O(3). This is a ball with unequal inertia coefficients rolling without slipping on the plane, with vertical rotations forbidden. Finally, we discuss reduction of internal symmetries. Chaplygin’s “marble,” where vertical rotations are allowed, is not Hamiltonizable at the compressed T*S O(3) level. We conjecture that it is also not Hamiltonizable when reduced to T*S 2.

Rubber rolling over a sphere

Abstract“Rubber” coated bodies rolling over a surface satisfy a no-twist condition in addition to the no slip condition satisfied by “marble” coated bodies [1]. Rubber rolling has an interesting differential geometric appeal because the geodesic curvatures of the curves on the surfaces at 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 exceptional group G2). The 2–3–5 nonholonomic geometries are classified in a companion paper [2] via Cartan’s equivalence method [3]. Rubber rolling of a convex body over a sphere defines a generalized Chaplygin system [4–8] with SO(3) symmetry group, total space Q = SO(3) × S2 and base S2, that can be reduced to an almost Hamiltonian system in T*S2 with a non-closed 2-form ωNH. In this paper we present some basic results on the sphere-sphere problem: a dynamically asymmetric but balanced sphere of radius b (unequal moments of inertia Ij but with center of gravity at the geometric center), rubber rolling over another sphere of radius a. In this example ωNH is conformally symplectic [9]: the reduced system becomes Hamiltonian after a coordinate dependent change of time. In particular there is an invariant measure, whose density is the determinant of the reduced Legendre transform, to the power p = 1/2(b/a − 1). Using sphero-conical coordinates we verify the result by Borisov and Mamaev [10] that the system is integrable for p = −1/2 (ball over a plane). They have found another integrable case [11] corresponding to p = −3/2 (rolling ball with twice the radius of a fixed internal ball). Strikingly, a different set of sphero-conical coordinates separates the Hamiltonian in this case. No other integrable cases with different Ij are known.

Moving frames for cotangent bundles

Abstract Cartans moving frames method is a standard tool in Riemannian geometry. We set up the machinery for applying moving frames to cotangent bundles and its sub-bundles defined by nonholonomic constraints.

Micro-swimming without flagella: Propulsion by internal structures

Since a first proof-of-concept for an autonomous micro-swimming device appeared in 2005 a strong interest on the subject ensued. The most common configuration consists of a cell driven by an external propeller, bio-inspired by bacteria such as E.coli. It is natural to investigate whether micro-robots powered by internal mechanisms could be competitive. We compute the translational and rotational velocity of a spheroid that produces a helical wave on its surface, as has been suggested for the rod-shaped cyanobacterium Synechococcus. This organisms swims up to ten body lengths per second without external flagella. For the mathematical analysis we employ the tangent plane approximation method, which is adequate for amplitudes, frequencies and wave lengths considered here. We also present a qualitative discussion about the efficiency of a device driven by an internal rotating structure.

Rubber Rolling: Geometry and Dynamics of 2-3-5 Distributions

We address two research lines, continuing our work in [11]. The first uses the affine connection introduced by Cartan at the 1928 International Congress of Mathematicians. We classify here the 2-3-5 nonholonomic geometries. The maximum symmetry case, 6-dimensional, has two branches. We describe the most interesting and quite surprising one, that ocurs in the celebrated 3:1 sphere-sphere distribution (a shadow of Cartan’s exceptional Lie group G2). In the second part we study the dynamics of a “rubber coated” body rolling without slipping nor twisting on a surface. If the latter is a sphere one has a SO(3) Chaplygin system [14], and the dynamics reduces to T* S2. The sphere-sphere problem is conformally symplectic. Details and further results will be published elsewhere (for the dynamic part, see [16]), and posted on Arxiv.

Self-propulsion of N-hinged ‘Animats’ at low reynolds number

Optimal locomotion of micro-organisms (on a small Reynolds number flow) can be regarded as a sub-riemannian geometry on a principal bundle with a mechanical connection. Aiming at robotic applications flagella are modeled as concatenated line segments with variable hinge angles. As an example, we consider E. Purcells 2-hinged animat [39], the simplest configuration capable to circumnvent Stokes flow reversibility.

On the dynamic Markov-Dubins problem: From path planning in robotics and biolocomotion to computational anatomy

Andrei Andreyevich Markov proposed in 1889 the problem (solved by Dubins in 1957) of finding the twice continuously differentiable (arc length parameterized) curve with bounded curvature, of minimum length, connecting two unit vectors at two arbitrary points in the plane. In this note we consider the following variant, which we call the dynamic Markov-Dubins problem (dM-D): to find the time-optimal C2 trajectory connecting two velocity vectors having possibly different norms. The control is given by a force whose norm is bounded. The acceleration may have a tangential component, and corners are allowed, provided the velocity vanishes there. We show that for almost all the two vectors boundary value conditions, the optimization problem has a smooth solution. We suggest some research directions for the dM-D problem on Riemannian manifolds, in particular we would like to know what happens if the underlying geodesic problem is completely integrable. Path planning in robotics and aviation should be the usual applications, and we suggest a pursuit problem in biolocomotion. Finally, we suggest a somewhat unexpected application to “dynamic imaging science”. Short time processes (in medicine and biology, in environment sciences, geophysics, even social sciences?) can be thought as tangent vectors. The time needed to connect two processes via a dynamic Markov-Dubins problem provides a notion of distance. Statistical methods could then be employed for classification purposes using a training set.

Non-Holonomic Systems with Symmetry Allowing a Conformally Symplectic Reduction

Non-holonomic mechanical systems can be described by a degenerate almostPoisson structure [10] (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 “noncompressed” constrained system never admits a Jacobi structure (in the sense of Marie et al. [4] [9]).