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Journal of Geometric Mechanics

2014 , Volume 6 , Issue 2

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Geometric characterization of the workspace of non-orthogonal rotation axes
Bertold Bongardt
2014, 6(2): 141-166 doi: 10.3934/jgm.2014.6.141 +[Abstract](281) +[PDF](8340.5KB)
In this article, a novel characterization of the workspace of 3R chains with non-orthogonal, intersecting axes is derived by describing the set of singular orientations as two tori that separate two-solvable from non-solvable orientations within $SO(3)$. Therefore, the tori provide the boundary of the workspace of the axes' constellation. The derived characterization generalizes a recent result obtained by Piovan and Bullo. It is based on a specific, novel representation of rotations, called unit ball representation, which allows to interpret the workspace characterization with ease. In an appendix, tools for dealing with angles and rotations are introduced and the equivalence of unit quaternion representation and unit ball representation is described.
An extension of the Dirac and Gotay-Nester theories of constraints for Dirac dynamical systems
Hernán Cendra, María Etchechoury and Sebastián J. Ferraro
2014, 6(2): 167-236 doi: 10.3934/jgm.2014.6.167 +[Abstract](420) +[PDF](922.8KB)
This paper extends the Gotay-Nester and the Dirac theories of constrained systems in order to deal with Dirac dynamical systems in the integrable case. Integrable Dirac dynamical systems are viewed as constrained systems where the constraint submanifolds are foliated. The cases considered usually in the literature correspond to a trivial foliation, with only one leaf. A Constraint Algorithm for Dirac dynamical systems (CAD), which extends the Gotay-Nester algorithm, is developed. Evolution equations are written using a Dirac bracket adapted to the foliations and an adapted total energy. The interesting example of LC circuits is developed in detail. The paper emphasizes the point of view that Dirac and Gotay-Nester theories are, in a certain sense, dual and that using a combination of results from both theories may have advantages in dealing with a given example, rather than using systematically one or the other.
Bifurcations of relative equilibria near zero momentum in Hamiltonian systems with spherical symmetry
James Montaldi
2014, 6(2): 237-260 doi: 10.3934/jgm.2014.6.237 +[Abstract](331) +[PDF](430.2KB)
For Hamiltonian systems with spherical symmetry there is a marked difference between zero and non-zero momentum values, and amongst all relative equilibria with zero momentum there is a marked difference between those of zero and those of non-zero angular velocity. We use techniques from singularity theory to study the family of relative equilibria that arise as a symmetric Hamiltonian which has a group orbit of equilibria with zero momentum is perturbed so that the zero-momentum relative equilibrium are no longer equilibria. We also analyze the stability of these perturbed relative equilibria, and consider an application to satellites controlled by means of rotors.
Periodic orbits in the Kepler-Heisenberg problem
Corey Shanbrom
2014, 6(2): 261-278 doi: 10.3934/jgm.2014.6.261 +[Abstract](283) +[PDF](498.1KB)
One can formulate the classical Kepler problem on the Heisenberg group, the simplest sub-Riemannian manifold. We take the sub-Riemannian Hamiltonian as our kinetic energy, and our potential is the fundamental solution to the Heisenberg sub-Laplacian. The resulting dynamical system is known to contain a fundamental integrable subsystem. Here we use variational methods to prove that the Kepler-Heisenberg system admits periodic orbits with $k$-fold rotational symmetry for any odd integer $k\geq 3$. Approximations are shown for $k=3$.

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