Two Examples of Integrable $G$-Strands

Details Two Examples of Integrable $G$-Strands Two Examples of Integrable $G$-Strands

2011-09-20

arxiv.org/abs/1109.4421v2

Nonlinear Sciences

Chaotic Dynamics

32 pages, 5 figures, 2nd version, comments welcome! New section 5.4 on G-(Diff)

Scientific paper

A $G$-strand is a map $g(t,x):\,\mathbb{R}\times\mathbb{R}\to G$ for a Lie group $G$ that follows from Hamilton's principle for a certain class of $G$-invariant Lagrangians. The SO(3)-strand is the $G$-strand version of the rigid body equation and it may be regarded physically as a continuous spin chain. Here, $SO(3)_K$-strand dynamics is derived as an Euler-Poincar\'e system for a certain class of variations and recast as a Lie-Poisson Hamiltonian system for coadjoint flow. For a special Hamiltonian, the $SO(3)_K$-strand for ellipsoidal rotations is mapped into a completely integrable generalization of the classical chiral model for the SO(3)-strand. Analogous results are obtained for the Sp(2)-strand. The Sp(2)-strand is the $G$-strand version of the $Sp(2)$ Bloch-Iserles ordinary differential equation, whose solutions exhibit dynamical sorting. Numerical solutions show nonlinear interactions of coherent wave-like solutions in both cases.

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