We consider a 3-dimensional elastic continuum whose material points
can experience no displacements, only rotations. This framework is a
special case of the Cosserat theory of elasticity. Rotations of
material points of the continuum are described mathematically by
attaching to each geometric point an orthonormal basis which gives a
field of orthonormal bases called the coframe. As the dynamical
variables (unknowns) of our theory we choose the coframe and a
density.
In the first part of the talk we write down the general dynamic
variational functional of our problem. In doing this we follow the
logic of classical linear elasticity with displacements replaced by
rotations and strain replaced by torsion. The corresponding
Euler-Lagrange equations turn out to be nonlinear, with the source
of this nonlinearity being purely geometric: unlike displacements,
rotations in 3D do not commute.
In the second part of the talk we present a class of explicit
solutions of our Euler-Lagrange equations. We call these solutions
plane waves. We identify two types of plane waves and calculate
their velocities.
In the third part of the talk we consider a particular case of our
theory when only one of the three rotational elastic moduli, that
corresponding to axial torsion, is nonzero. We examine this case in
detail and seek solutions which oscillate harmonically in time but
depend on the space coordinates in an arbitrary manner (this is a
far more general setting than with plane waves). We show [1] that
our second order nonlinear Euler-Lagrange equations are equivalent
to a pair of linear first order massless Dirac equations. The
crucial element of the proof is the observation that our Lagrangian
admits a factorisation.
[1] Olga Chervova and Dmitri Vassiliev, "The stationary Weyl
equation and Cosserat elasticity", preprint http://arxiv.org/abs/1001.4726