Abstract
A G-strand is a map g(t, s) : ℝ × ℝ→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-Poincaré 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).
Original language | English |
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Pages (from-to) | 517-551 |
Number of pages | 35 |
Journal | Journal of Nonlinear Science |
Volume | 22 |
Issue number | 4 |
DOIs | |
Publication status | Published - Aug 2012 |
Keywords
- Bloch-Iserles equation
- Chiral models
- Euler-Poincaré equations
- Hamilton's principle
- Integrable Hamiltonian systems
- Inverse Spectral Transform (IST)
- Momentum maps
- Sobolev norms
- Solitons
- Spin chains