$G$-Strands

Abstract

A $G$-strand is a map $g(t,s):\mathbb{R}\times \mathbb{R}\to G$ 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 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. $\mathrm{Diff}(\mathbb{R})$-strand equations on the diffeomorphism group $G=\mathrm{Diff}(\mathbb{R})$ are also introduced and shown to admit solutions with singular support (e.g., peakons).

35 pages, 5 figures, 3rd version. To appear in J Nonlin Sci

References in corpus (4)

• Vlasov moments, integrable systems and singular solutions
• Vlasov moment flows and geodesics on the Jacobi group
• Dynamics of charged molecular strands
• Hybrid models for complex fluids with multipolar interactions

Cited by in corpus (14)

• Inexact trajectory planning and inverse problems in the Hamilton--Pontryagin framework
• Generating Functionals and Lagrangian PDEs
• Geometric integrators for higher-order mechanics on Lie groups
• Matrix G-Strands
• Hybrid models for complex fluids with multipolar interactions
• $G$-Strands on symmetric spaces
• Poisson-Poincaré reduction for Field Theories
• Clebsch variational principles in field theories and singular solutions of covariant EPDiff equations
• Integrable G-Strands on semisimple Lie groups
• $G$-Strands and Peakon Collisions on $\mathrm{Diff}(\mathbb{R})$
• Toda lattice G-Strands
• Euler-Poincaré equations for $G$-Strands
• A distance on curves modulo rigid transformations
• Hamiltonian Reduction in Affine Principal Bundles