This paper studies the stability and bifurcations of the relative equilibria of the double spherical pendulum, which has the circle as its symmetry group. This example as well as others with nonabelian symmetry groups, such as the rigid body, illustrate some useful general theory about Lagrangian reduction. In particular, we establish a satisfactory global theory of Lagrangian reduction that is...

We develop a new cell-centered control volume Lagrangian scheme for solving Euler equations of compressible gas dynamics in cylindrical coordinates. Based on a local coordinate transform strategy, the scheme can preserve one-dimensional spherical symmetry in a two-dimensional cylindrical geometry when computed on an equal-angle-zoned initial grid. Unlike many previous area weighted schemes that...

Organization of spherical particles into lattices is typically driven by packing considerations. Although the addition of directional binding can significantly broaden structural diversity, nanoscale implementation remains challenging. Here we investigate the assembly of clusters and lattices in which anisotropic polyhedral blocks coordinate isotropic spherical nanoparticles via shape-induced d...

In this paper, we implement finite volume Weighted Essentially Non-Oscillatory (WENO) schemes in both cylindrical and spherical coordinate systems for the Euler equations with cylindrical or spherical symmetry. We analyze three different spatial discretizations: one that is shown to be high-order accurate but not conservative, one conservative but not high-order accurate, and one both high-orde...