Determining 3-D Motion and Structure of a Rigid Body Using the Spherical Projection

Coordinated Science Laboratory was formerly known as Control Systems LaboratoryNational Science Foundation / NSF ECS 81-1208

Standoff Distance of Bow Shocks in Galaxy Clusters as Proxy for Mach Number

X-ray observations of merging clusters provide many examples of bow shocks leading merging subclusters. While the Mach number of a shock can be estimated from the observed density jump using Rankine-Hugoniot condition, it reflects only the velocity of the shock itself and is generally not equal to the velocity of the infalling subcluster dark matter halo or to the velocity of the contact discontinuity separating gaseous atmospheres of the two subclusters. Here we systematically analyze additional information that can be obtained by measuring the standoff distance, i.e. the distance between the leading edge of the shock and the contact discontinuity that drives this shock. The standoff distance is influenced by a number of additional effects, e.g. (1) the gravitational pull of the main cluster (causing acceleration/deceleration of the infalling subcluster), (2) the density and pressure gradients of the atmosphere in the main cluster, (3) the non-spherical shape of the subcluster, and (4) projection effects. The first two effects tend to bias the standoff distance in the same direction, pushing the bow shock closer to (farther away from) the subcluster during the pre- (post-)merger stages. Particularly, in the post-merger stage, the shock could be much farther away from the subcluster than predicted by a model of a body moving at a constant speed in a uniform medium. This implies that a combination of the standoff distance with measurements of the Mach number from density/temperature jumps can provide important information on the merger, e.g. differentiating between the pre- and post-merger stages.Comment: 11 pages, 12 figures. Including major revision and matched to accepted version in MNRA

Rigid body trajectories in different 6D spaces

The objective of this paper is to show that the group SE(3) with an imposed Lie-Poisson structure can be used to determine the trajectory in a spatial frame of a rigid body in Euclidean space. Identical results for the trajectory are obtained in spherical and hyperbolic space by scaling the linear displacements appropriately, since the influence of the moments of inertia on the trajectories tend to zero as the scaling factor increases. The semi-direct product of the linear and rotational motions gives the trajectory from a body frame perspective. It is shown that this cannot be used to determine the trajectory in the spatial frame. The body frame trajectory is thus independent of the velocity coupling. In addition, it is shown that the analysis can be greatly simplified by aligning the axes of the spatial frame with the axis of symmetry which is unchanging for a natural system with no forces and rotation about an axis of symmetry

Relative Critical Points

Relative equilibria of Lagrangian and Hamiltonian systems with symmetry are critical points of appropriate scalar functions parametrized by the Lie algebra (or its dual) of the symmetry group. Setting aside the structures - symplectic, Poisson, or variational - generating dynamical systems from such functions highlights the common features of their construction and analysis, and supports the construction of analogous functions in non-Hamiltonian settings. If the symmetry group is nonabelian, the functions are invariant only with respect to the isotropy subgroup of the given parameter value. Replacing the parametrized family of functions with a single function on the product manifold and extending the action using the (co)adjoint action on the algebra or its dual yields a fully invariant function. An invariant map can be used to reverse the usual perspective: rather than selecting a parametrized family of functions and finding their critical points, conditions under which functions will be critical on specific orbits, typically distinguished by isotropy class, can be derived. This strategy is illustrated using several well-known mechanical systems - the Lagrange top, the double spherical pendulum, the free rigid body, and the Riemann ellipsoids - and generalizations of these systems

Hamiltonian G-Spaces with Regular Momenta

Let G be a compact connected non-Abelian Lie group and let (P, w, G, J) be a Hamiltonian G-space. Call this space a G-space with regular momenta if J(P) ⊂ g*reg, here g*reg⊂g* denotes the regular points of the co-adjoint action of G. Here problems involving a G-space with regular momenta are reduced to problems in an associated lower dimensional Hamiltonian T-space, where T ⊂ G is a maximal torus. For example two such G-spaces are shown to be equivalent if and only if they have equivalent associated T-spaces. We also give a new construction of a normal form due to Marle (1983), for integrable G-spaces with regular momenta. We show that this construction, which is a kind of non-Abelian generalization of action-angle coordinates, can be reduced to constructing conventional action-angle coordinates in the associated T-space. In particular the normal form applies globally if the action-angle coordinates can be constructed globally. We illustrate our results in concrete examples from mechanics, including the rigid body. We also indicate applications to Hamiltonian perturbation theory

Systems of Hess-Appel'rot Type and Zhukovskii Property

We start with a review of a class of systems with invariant relations, so called {\it systems of Hess--Appel'rot type} that generalizes the classical Hess--Appel'rot rigid body case. The systems of Hess-Appel'rot type carry an interesting combination of both integrable and non-integrable properties. Further, following integrable line, we study partial reductions and systems having what we call the {\it Zhukovskii property}: these are Hamiltonian systems with invariant relations, such that partially reduced systems are completely integrable. We prove that the Zhukovskii property is a quite general characteristic of systems of Hess-Appel'rote type. The partial reduction neglects the most interesting and challenging part of the dynamics of the systems of Hess-Appel'rot type - the non-integrable part, some analysis of which may be seen as a reconstruction problem. We show that an integrable system, the magnetic pendulum on the oriented Grassmannian $Gr^+(4,2)$ has natural interpretation within Zhukovskii property and it is equivalent to a partial reduction of certain system of Hess-Appel'rot type. We perform a classical and an algebro-geometric integration of the system, as an example of an isoholomorphic system. The paper presents a lot of examples of systems of Hess-Appel'rot type, giving an additional argument in favor of further study of this class of systems.Comment: 42 page

The inelastic hard dimer gas: a non-spherical model for granular matter

We study a two-dimensional gas of inelastic smooth hard dimers. Since the collisions between dimers are dissipative, being characterized by a coefficient of restitution $\alpha<1$, and no external driving force is present, the energy of the system decreases in time and no stationary state is achieved. However, the resulting non equilibrium state of the system displays several interesting properties in close analogy with systems of inelastic hard spheres, whose relaxational dynamics has been thoroughly explored. We generalise to inelastic systems a recently method introduced [G.Ciccotti and G.Kalibaeva, J. Stat. Phys. {\bf 115}, 701 (2004)] to study the dynamics of rigid elastic bodies made up of different spheres hold together by rigid bonds. Each dimer consists of two hard disks of diameter $d$, whose centers are separated by a fixed distance $a$. By describing the rigid bonds by means of holonomic constraints and deriving the appropriate collision rules between dimers, we reduce the dynamics to a set of equations which can be solved by means of event driven simulation. After deriving the algorithm we study the decay of the total kinetic energy, and of the ratio between the rotational and the translational kinetic energy of inelastic dimers. We show numerically that the celebrated Haff's homogeneous cooling law $t^{-2}$, describing how the kinetic energy of an inelastic hard sphere system with constant coefficient of restitution decreases in time, holds even in the case of these non spherical particles. We fully characterize this homogeneous decay process in terms of appropriate decay constants and confirm numerically the scaling behavior of the velocity distributions.Comment: 21 pages, 6 figures and 2 tables, submitted to JC