G-Strands

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).

Invariant variational problems on homogeneous spaces

“…the principal chiral model [14]. A preferred form for the reduced Lagrangian ℓ : g × g → R, both for strands on matrix groups and for strands on diffeomorphism groups, is ℓ(ξ , η) = 1 2 |ξ | 2 − 1 2 |η| 2 .…”

G -Strands on symmetric spaces

“…Despite the complete integrability of this system, we observe instabilities for high frequencies. This is found from the dispersion relation of the linearized equation around the equilibrium solution X e = xe 1 , Y e = ye 1 and Z e = ze 1 , with x, y, x ∈ R, by…”

“…We display in figure 2 a few snapshots of the collision, obtain by plotting (4.31) with the harmonic function (4.32) with k = 3. 1 …”

“…We study a simplified field theoretical model called G-strands in which the fields take values in a Lie group G. The G-strands are related to the chiral model of particle physics, and they have been studied in the context of geometric mechanics in [1][2][3][4][5]. In this paper, we are considering G-strands where the Lie group is replaced by a symmetric space.…”

“…In our previous publications [1][2][3][4][5], we introduced and studied the G-strand construction, which gives rise to equations for a map R × R into a Lie group G associated with a G-invariant Lagrangian. In the case of a semisimple Lie group G with a Lie algebra g, various classes of integrable equations have been found.…”