5 research outputs found
Differential Geometry applied to Acoustics : Non Linear Propagation in Reissner Beams
Although acoustics is one of the disciplines of mechanics, its
"geometrization" is still limited to a few areas. As shown in the work on
nonlinear propagation in Reissner beams, it seems that an interpretation of the
theories of acoustics through the concepts of differential geometry can help to
address the non-linear phenomena in their intrinsic qualities. This results in
a field of research aimed at establishing and solving dynamic models purged of
any artificial nonlinearity by taking advantage of symmetry properties
underlying the use of Lie groups. The geometric constructions needed for
reduction are presented in the context of the "covariant" approach.Comment: Submitted to GSI2013 - Geometric Science of Informatio
-Strands
A -strand is a map for a Lie
group that follows from Hamilton's principle for a certain class of
-invariant Lagrangians. The SO(3)-strand is the -strand version of the
rigid body equation and it may be regarded physically as a continuous spin
chain. Here, -strand dynamics for ellipsoidal rotations is derived as
an Euler-Poincar\'e system for a certain class of variations and recast as a
Lie-Poisson system for coadjoint flow with the same Hamiltonian structure as
for a perfect complex fluid. For a special Hamiltonian, the -strand is
mapped into a completely integrable generalization of the classical chiral
model for the SO(3)-strand. Analogous results are obtained for the
-strand. The -strand is the -strand version of the
Bloch-Iserles ordinary differential equation, whose solutions exhibit dynamical
sorting. Numerical solutions show nonlinear interactions of coherent wave-like
solutions in both cases. -strand equations on the
diffeomorphism group are also introduced and shown
to admit solutions with singular support (e.g., peakons).Comment: 35 pages, 5 figures, 3rd version. To appear in J Nonlin Sc
Euler-Poincar\'e approaches to nematodynamics
Nematodynamics is the orientation dynamics of flowless liquid-crystals. We
show how Euler-Poincar\'e reduction produces a unifying framework for various
theories, including Ericksen-Leslie, Luhiller-Rey, and Eringen's micropolar
theory. In particular, we show that these theories are all compatible with each
other and some of them allow for more general configurations involving a non
vanishing discination density. All results are also extended to flowing liquid
crystals.Comment: 26 pages, no figure