Kurt M. Ehlers

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.

Blue moons and Martian sunsets

The familiar yellow or orange disks of the moon and sun, especially when they are low in the sky, and brilliant red sunsets are a result of the selective extinction (scattering plus absorption) of blue light by atmospheric gas molecules and small aerosols, a phenomenon explainable using the Rayleigh scattering approximation. On rare occasions, dust or smoke aerosols can cause the extinction of red light to exceed that for blue, resulting in the disks of the sun and moon to appear as blue. Unlike Earth, the atmosphere of Mars is dominated by micron-size dust aerosols, and the sky during sunset takes on a bluish glow. Here we investigate the role of dust aerosols in the blue Martian sunsets and the occasional blue moons and suns on Earth. We use the Mie theory and the Debye series to calculate the wavelength-dependent optical properties of dust aerosols most commonly found on Mars. Our findings show that while wavelength selective extinction can cause the suns disk to appear blue, the color of the glow surrounding the sun as observed from Mars is due to the dominance of near-forward scattering of blue light by dust particles and cannot be explained by a simple, Rayleigh-like selective extinction explanation.