Point singularities on 2D surfaces

2013-06-28This work studies point vortices on a sphere and complex point singularities on a plane. The motivation for the study is to get deeper understanding of the dynamics of symmetric configurations of point vortices and point singularities. Equations of point vortex motion are derived from the Euler equations. Geometric description of the phase space is given along with symplectic structure and Lie-Poisson brackets. Symplectic reduction is performed and reduced Hamiltonian is found. Configuration matrix approach is used to find fixed equilibrium configurations of point singularities and relative equilibrium configurations of point vortices. Based on this method, relative equilibria in the form of tetrahedron, octahedron, cube, icosahedron, dodecahedron are described. Using energy-momentum method conducted study of stability of general tetrahedral and octahedral configurations. For the cubic, icosahedral and dodecahedral cases studied stability of superpositions of axis-symmetric configurations. For the tetrahedral, cubic and icosahedral configuration regions of stability are plotted. Instability results for special cases of cubic and icosahedral configurations are proved. ❧ Fixed equilibrium configurations of point singularities on a plane are found. Theorems about existence and uniqueness of the equilibria are proved. For each of the configuration, singular value decomposition is performed. The singular values are used to obtain probability distribution and Shannon entropy for the configurations is computed. Relative equilibria for even and odd number of point singularities are described. Relative equilibria for 2, 3, 4 point singularities are studied. For higher number of singularities method of finding relative equilibria is provided