419 research outputs found
-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
Integrable G-Strands on semisimple Lie groups
The present paper derives systems of partial differential equations that
admit a quadratic zero curvature representation for an arbitrary real
semisimple Lie algebra. It also determines the general form of Hamilton's
principles and Hamiltonians for these systems and analyzes the linear stability
of their equilibrium solutions in the examples of and
.Comment: 17 pages, no figures. First version, comments welcome
Matrix G-strands
We discuss three examples in which one may extend integrable Euler–Poincare ordinary differential equations to integrable Euler–Poincare partial differentialequations in the matrix G-Strand context. After describing matrix G-Strand examples for SO(3) and SO(4) we turn our attention to SE(3) where the matrix G-Strand equations recover the exact rod theory in the convective representation. We then find a zero curvature representation of these equations and establish the conditions under which they are completely integrable. Thus, the G-Strand equations turn out to be a rich source of integrable systems. The treatment is meant to be expository and most concepts are explained in examples in the language of vectors in R3
Euler-Poincar\'e equations for -Strands
The -strand equations for a map into a Lie
group are associated to a -invariant Lagrangian. The Lie group manifold
is also the configuration space for the Lagrangian. The -strand itself is
the map , where and are the
independent variables of the -strand equations. The Euler-Poincar\'e
reduction of the variational principle leads to a formulation where the
dependent variables of the -strand equations take values in the
corresponding Lie algebra and its co-algebra,
with respect to the pairing provided by the variational derivatives of the
Lagrangian.
We review examples of different -strand constructions, including matrix
Lie groups and diffeomorphism group. In some cases the -strand equations are
completely integrable 1+1 Hamiltonian systems that admit soliton solutions.Comment: To appear in Conference Proceedings for Physics and Mathematics of
Nonlinear Phenomena, 22 - 29 June 2013, Gallipoli (Italy)
http://pmnp2013.dmf.unisalento.it/talks.shtml, 9 pages, no figure
-Strands and Peakon Collisions on
A -strand is a map for a Lie group
that follows from Hamilton's principle for a certain class of -invariant
Lagrangians. Some -strands on finite-dimensional groups satisfy 1+1
space-time evolutionary equations that admit soliton solutions as completely
integrable Hamiltonian systems. For example, the -strand equations
may be regarded physically as integrable dynamics for solitons on a continuous
spin chain. Previous work has shown that -strands for diffeomorphisms on the
real line possess solutions with singular support (e.g. peakons). This paper
studies collisions of such singular solutions of -strands when is the group of diffeomorphisms of the real line
, for which the group product is composition of smooth invertible
functions. In the case of peakon-antipeakon collisions, the solution reduces to
solving either Laplace's equation or the wave equation (depending on a sign in
the Lagrangian) and is written in terms of their solutions. We also consider
the complexified systems of -strand equations for
corresponding to a harmonic map and
find explicit expressions for its peakon-antipeakon solutions, as well.Comment: arXiv:1109.4421 introduced singular solutions of G-strand equations
on the diffeos. This paper solves the equations for their pairwise
interactio
G-Strands on symmetric spaces.
We study the G-strand equations that are extensions of the classical chiral model of particle physics in the particular setting of broken symmetries described by symmetric spaces. These equations are simple field theory models whose configuration space is a Lie group, or in this case a symmetric space. In this class of systems, we derive several models that are completely integrable on finite dimensional Lie group G, and we treat in more detail examples with symmetric space SU(2)/S(1) and SO(4)/SO(3). The latter model simplifies to an apparently new integrable nine-dimensional system. We also study the G-strands on the infinite dimensional group of diffeomorphisms, which gives, together with the Sobolev norm, systems of 1+2 Camassa-Holm equations. The solutions of these equations on the complementary space related to the Witt algebra decomposition are the odd function solutions
G-Strands on Symmetric Spaces
We study the G-strand equations that are extensions of the classical chiral model of particle physics in the particular setting of broken symmetries described by symmetric spaces. These equations are simple field theory models whose configuration space is a Lie group, or in this case a symmetric space. In this class of systems, we derive several models that are completely integrable on finite dimensional Lie group G, and we treat in more detail examples with symmetric space SU(2)/S1 and SO(4)/SO(3). The latter model simplifies to an apparently new integrable nine-dimensional system. We also study the G-strands on the infinite dimensional group of diffeomorphisms, which gives, together with the Sobolev norm, systems of 1+2 Camassa–Holm equations. The solutions of these equations on the complementary space related to the Witt algebra decomposition are the odd function solutions
G-Strands and Peakon Collisions on Diff(R)
A G-strand is a map g : R x R --\u3e G for a Lie group G that follows from Hamilton\u27s principle for a certain class of G-invariant Lagrangians. Some G-strands on finite-dimensional groups satisfy 1+1 space-time evolutionary equations that admit soliton solutions as completely integrable Hamiltonian systems. For example, the SO(3)-strand equations may be regarded physically as integrable dynamics for solitons on a continuous spin chain. Previous work has shown that G-strands for diffeomorphisms on the real line possess solutions with singular support (e.g. peakons). This paper studies collisions of such singular solutions of G-strands when G = Diff(R) is the group of diffeomorphisms of the real line R, for which the group product is composition of smooth invertible functions. In the case of peakon-antipeakon collisions, the solution reduces to solving either Laplace\u27s equation or the wave equation (depending on a sign in the Lagrangian) and is written in terms of their solutions. We also consider the complexified systems of G-strand equations for G = Diff(R) corresponding to a harmonic map g : C --\u3e Diff(R) and find explicit expressions for its peakon-antipeakon solutions, as well
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