Kurt Ehlers
7000 Dandino Boulevard
Reno, NV, 895123999, USA
Mathematics Department,
Truckee Meadows Community College
Publications:
Ehlers K. M., Koiller J.
Microswimming Without Flagella: Propulsion by Internal Structures
2011, vol. 16, no. 6, pp. 623652
Abstract
Since a first proofofconcept for an autonomous microswimming device appeared in 2005 a strong interest on the subject ensued. The most common configuration consists of a cell driven by an external propeller, bioinspired by bacteria such as E.coli. It is natural to investigate whether microrobots 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 rodshaped 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.

Koiller J., Ehlers K. M.
Rubber Rolling over a Sphere
2007, vol. 12, no. 2, pp. 127152
Abstract
"Rubber" coated bodies rolling over a surface satisfy a notwist 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 235 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 $G_2$). The 235 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 [48] with $SO$(3) symmetry group, total space $Q = SO(3) \times S^2$ and base $S^2$, that can be reduced to an almost Hamiltonian system in $T^*S^2$ with a nonclosed 2form $\omega_{NH}$. In this paper we present some basic results on the spheresphere problem: a dynamically asymmetric but balanced sphere of radius $b$ (unequal moments of inertia $I_j$ but with center of gravity at the geometric center), rubber rolling over another sphere of radius $a$. In this example $\omega_{NH}$ 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 spheroconical 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 spheroconical coordinates separates the Hamiltonian in this case. No other integrable cases with different $I_j$ are known.
