Clumpiness in Star Forming Regions

In this article, some aspects of the clumpy nature of molecular clouds are reviewed. In particular the observational evidence for small-scale structures both in low and high mass star forming regions will be discussed. I will review some examples of `clumpiness' such as: i) the molecular clumps ahead of HH objects and how the study of the physical and chemical nature of these clumps is important for the understanding of the clumpiness of the Interstellar Medium; and ii)hot cores and their use as a tool to study the early phases of massive star formation. This revised version was published online in July 2006 with corrections to the Cover Date.     

Geometric Derivation of the Delaunay Variables and Geometric Phases

We derive the classical Delaunay variables by finding a suitable symmetry action of the three torus T³ on the phase space of the Kepler problem, computing its associated momentum map and using the geometry associated with this structure. A central feature in this derivation is the identification of the mean anomaly as the angle variable for a symplectic S ¹ action on the union of the non-degenerate elliptic Kepler orbits. This approach is geometrically more natural than traditional ones such as directly solving Hamilton–Jacobi equations, or employing the Lagrange bracket. As an application of the new derivation, we give a singularity free treatment of the averaged J ₂-dynamics (the effect of the bulge of the Earth) in the Cartesian coordinates by making use of the fact that the averaged J ₂-Hamiltonian is a collective Hamiltonian of the T³ momentum map. We also use this geometric structure to identify the drifts in satellite orbits due to the J ₂ effect as geometric phases.