Hamiltonian Frenet-Serret dynamics

The Hamiltonian formulation of the dynamics of a relativistic particle described by a higher-derivative action that depends both on the first and the second Frenet-Serret curvatures is considered from a geometrical perspective. We demonstrate how reparametrization covariant dynamical variables and their projections onto the Frenet-Serret frame can be exploited to provide not only a significant simplification of but also novel insights into the canonical analysis. The constraint algebra and the Hamiltonian equations of motion are written down and a geometrical interpretation is provided for the canonical variables.Comment: Latex file, 14 pages, no figures. Revised version to appear in Class. Quant. Gra

Axially symmetric membranes with polar tethers

Axially symmetric equilibrium configurations of the conformally invariant Willmore energy are shown to satisfy an equation that is two orders lower in derivatives of the embedding functions than the equilibrium shape equation, not one as would be expected on the basis of axial symmetry. Modulo a translation along the axis, this equation involves a single free parameter c.If c\ne 0, a geometry with spherical topology will possess curvature singularities at its poles. The physical origin of the singularity is identified by examining the Noether charge associated with the translational invariance of the energy; it is consistent with an external axial force acting at the poles. A one-parameter family of exact solutions displaying a discocyte to stomatocyte transition is described.Comment: 13 pages, extended and revised version of Non-local sine-Gordon equation for the shape of axi-symmetric membrane

Deformations of extended objects with edges

We present a manifestly gauge covariant description of fluctuations of a relativistic extended object described by the Dirac-Nambu-Goto action with Dirac-Nambu-Goto loaded edges about a given classical solution. Whereas physical fluctuations of the bulk lie normal to its worldsheet, those on the edge possess an additional component directed into the bulk. These fluctuations couple in a non-trivial way involving the underlying geometrical structures associated with the worldsheet of the object and of its edge. We illustrate the formalism using as an example a string with massive point particles attached to its ends.Comment: 17 pages, revtex, to appear in Phys. Rev. D5

Covariant perturbations of domain walls in curved spacetime

A manifestly covariant equation is derived to describe the perturbations in a domain wall on a given background spacetime. This generalizes recent work on domain walls in Minkowski space and introduces a framework for examining the stability of relativistic bubbles in curved spacetimes.Comment: 15 pages,ICN-UNAM-93-0

Open strings with topologically inspired boundary conditions

We consider an open string described by an action of the Dirac-Nambu-Goto type with topological corrections which affect the boundary conditions but not the equations of motion. The most general addition of this kind is a sum of the Gauss-Bonnet action and the first Chern number (when the background spacetime dimension is four) of the normal bundle to the string worldsheet. We examine the modification introduced by such terms in the boundary conditions at the ends of the string.Comment: 12 pages, late

Hamilton's equations for a fluid membrane: axial symmetry

Consider a homogenous fluid membrane, or vesicle, described by the Helfrich-Canham energy, quadratic in the mean curvature. When the membrane is axially symmetric, this energy can be viewed as an `action' describing the motion of a particle; the contours of equilibrium geometries are identified with particle trajectories. A novel Hamiltonian formulation of the problem is presented which exhibits the following two features: {\it (i)} the second derivatives appearing in the action through the mean curvature are accommodated in a natural phase space; {\it (ii)} the intrinsic freedom associated with the choice of evolution parameter along the contour is preserved. As a result, the phase space involves momenta conjugate not only to the particle position but also to its velocity, and there are constraints on the phase space variables. This formulation provides the groundwork for a field theoretical generalization to arbitrary configurations, with the particle replaced by a loop in space.Comment: 11 page

A New Anomaly-Free Gauged Supergravity in Six Dimensions

We present a new anomaly-free gauged N=1 supergravity model in six dimensions. The gauge group is E_7xG_2xU(1)_R, with all hyperinos transforming in the product representation {56,14). The theory admits monopole compactifications to R^4xS^2, leading to D=4 effective theories with broken supersymmetry and massless fermions.Comment: 9 pages, RevTeX

Force dipoles and stable local defects on fluid vesicles

An exact description is provided of an almost spherical fluid vesicle with a fixed area and a fixed enclosed volume locally deformed by external normal forces bringing two nearby points on the surface together symmetrically. The conformal invariance of the two-dimensional bending energy is used to identify the distribution of energy as well as the stress established in the vesicle. While these states are local minima of the energy, this energy is degenerate; there is a zero mode in the energy fluctuation spectrum, associated with area and volume preserving conformal transformations, which breaks the symmetry between the two points. The volume constraint fixes the distance $S$, measured along the surface, between the two points; if it is relaxed, a second zero mode appears, reflecting the independence of the energy on $S$; in the absence of this constraint a pathway opens for the membrane to slip out of the defect. Logarithmic curvature singularities in the surface geometry at the points of contact signal the presence of external forces. The magnitude of these forces varies inversely with $S$ and so diverges as the points merge; the corresponding torques vanish in these defects. The geometry behaves near each of the singularities as a biharmonic monopole, in the region between them as a surface of constant mean curvature, and in distant regions as a biharmonic quadrupole. Comparison of the distribution of stress with the quadratic approximation in the height functions points to shortcomings of the latter representation. Radial tension is accompanied by lateral compression, both near the singularities and far away, with a crossover from tension to compression occurring in the region between them.Comment: 26 pages, 10 figure

Helfrich-Canham bending energy as a constrained non-linear sigma model

The Helfrich-Canham bending energy is identified with a non-linear sigma model for a unit vector. The identification, however, is dependent on one additional constraint: that the unit vector be constrained to lie orthogonal to the surface. The presence of this constraint adds a source to the divergence of the stress tensor for this vector so that it is not conserved. The stress tensor which is conserved is identified and its conservation shown to reproduce the correct shape equation.Comment: 5 page

Chern-Simons theory and three-dimensional surfaces

There are two natural Chern-Simons theories associated with the embedding of a three-dimensional surface in Euclidean space; one is constructed using the induced metric connection -- it involves only the intrinsic geometry, the other is extrinsic and uses the connection associated with the gauging of normal rotations. As such, the two theories appear to describe very different aspects of the surface geometry. Remarkably, at a classical level, they are equivalent. In particular, it will be shown that their stress tensors differ only by a null contribution. Their Euler-Lagrange equations provide identical constraints on the normal curvature. A new identity for the Cotton tensor is associated with the triviality of the Chern-Simons theory for embedded hypersurfaces implied by this equivalence. The corresponding null surface stress capturing this information will be constructed explicitly.Comment: 10 pages, unnecessary details removed, typos fixed, references adde