103 research outputs found
Comment on "Classical Mechanics of Nonconservative Systems"
A Comment on the Letter by C. R. Galley, Phys. Rev. Lett. 110, 174301 (2013)
Chain Paradoxes
For nearly two centuries the dynamics of chains have offered examples of
paradoxical theoretical predictions. Here we propose a theory for the
dissipative dynamics of one-dimensional continua with singularities which
provides a unified treatment for chain problems that have suffered from
paradoxical solutions. These problems are duly solved within the present theory
and their paradoxes removed---we hope
Dissipative Shocks behind Bacteria Gliding
Gliding is a means of locomotion on rigid substrates utilized by a number of
bacteria includingmyxobacteria and cyanobacteria. One of the hypotheses
advanced to explain this motility mechanism hinges on the role played by the
slime filaments continuously extruded from gliding bacteria. This paper solves
in full a non-linear mechanical theory that treats as dissipative shocks both
the point where the extruded slime filament comes in contact with the
substrate, called the filament's foot, and the pore on the bacterium outer
surface from where the filament is ejected. We prove that kinematic
compatibility for shock propagation requires that the bacterium uniform gliding
velocity (relative to the substrate) and the slime ejecting velocity (relative
to the bacterium) must be equal, a coincidence that seems to have already been
observed.Comment: arXiv admin note: text overlap with arXiv:1402.636
Dissipative shocks in a chain fountain
The fascinating and anomalous behaviour of a chain that instead of falling
straight down under gravity, first rises and then falls, acquiring a steady
shape in space that resembles a fountain's sprinkle, has recently attracted
both popular and academic interest. The paper presents a complete mathematical
solution of this problem, whose distinctive feature is the introduction of a
number of dissipative shocks which can be resolved exactly
Octupolar order in two dimensions
Octupolar order is described in two space dimensions in terms of the maxima
(and conjugated minima) of the probability density associated with a
third-rank, fully symmetric and traceless tensor. Such a representation is
shown to be equivalent to diagonalizing the relevant third-rank tensor, an
equivalence which however is only valid in the two-dimensional case
The symmetries of octupolar tensors
Octupolar tensors are third order, completely symmetric and traceless
tensors. Whereas in 2D an octupolar tensor has the same symmetries as an
equilateral triangle and can ultimately be identified with a vector in the
plane, the symmetries that it enjoys in 3D are quite different, and only
exceptionally reduce to those of a regular tetrahedron. By use of the octupolar
potential that is, the cubic form associated on the unit sphere with an
octupolar tensor, we shall classify all inequivalent octupolar symmetries. This
is a mathematical study which also reviews and incorporates some previous, less
systematic attempts
Explicit excluded volume of cylindrically symmetric convex bodies
We represent explicitly the excluded volume Ve{B1,B2} of two generic
cylindrically symmetric, convex rigid bodies, B1 and B2, in terms of a family
of shape functionals evaluated separately on B1 and B2. We show that Ve{B1,B2}
fails systematically to feature a dipolar component, thus making illusory the
assignment of any shape dipole to a tapered body in this class. The method
proposed here is applied to cones and validated by a shape-reconstruction
algorithm. It is further applied to spheroids (ellipsoids of revolution), for
which it shows how some analytic estimates already regarded as classics should
indeed be emended
Octupolar Tensors for Liquid Crystals
A third-order three-dimensional symmetric traceless tensor, called the
\emph{octupolar} tensor, has been introduced to study tetrahedratic nematic
phases in liquid crystals. The octupolar \emph{potential}, a scalar-valued
function generated on the unit sphere by that tensor, should ideally have four
maxima capturing the most probable molecular orientations (on the vertices of a
tetrahedron), but it was recently found to possess an equally generic variant
with \emph{three} maxima instead of four. It was also shown that the
irreducible admissible region for the octupolar tensor in a three-dimensional
parameter space is bounded by a dome-shaped surface, beneath which is a
\emph{separatrix} surface connecting the two generic octupolar states. The
latter surface, which was obtained through numerical continuation, may be
physically interpreted as marking a possible \emph{intra-octupolar} transition.
In this paper, by using the resultant theory of algebraic geometry and the
E-characteristic polynomial of spectral theory of tensors, we give a
closed-form, algebraic expression for both the dome-shaped surface and the
separatrix surface. This turns the envisaged intra-octupolar transition into a
quantitative, possibly observable prediction. Some other properties of
octupolar tensors are also studied
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