Document Type
Article
Rights
This item is available under a Creative Commons License for non-commercial use only
Disciplines
Applied mathematics
Abstract
A G-strand is a map g(t,s): RxR --> 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 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 SO(3)K-strand 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. Diff(R)-strand equations on the diffeomorphism group G=Diff(R) are also introduced and shown to admit solutions with singular support (e.g., peakons).
DOI
http://doi.org10.21427/p8x2-jf40
Recommended Citation
D. D. Holm, R. I. Ivanov and J.R. Percival, G-Strands, J. Nonlinear Sciences, 22 (2012) pp 517–551. doi :10.21427/p8x2-jf40
Included in
Dynamical Systems Commons, Non-linear Dynamics Commons, Ordinary Differential Equations and Applied Dynamics Commons, Partial Differential Equations Commons
Publication Details
Journal of Nonlinear Sciences, 22 (2012) pp 517–551.