Related papers: $G$-Strands

A $G$-strand is a map $g:\mathbb{R}\times\mathbb{R}\to G$ for a Lie group $G$ that follows from Hamilton's principle for a certain class of $G$-invariant Lagrangians. Some $G$-strands on finite-dimensional groups satisfy 1+1 space-time…

The $G$-strand equations for a map $\mathbb{R}\times \mathbb{R}$ into a Lie group $G$ are associated to a $G$-invariant Lagrangian. The Lie group manifold is also the configuration space for the Lagrangian. The $G$-strand itself is the map…

We study the $G$-strand equations that are extensions of the classical chiral model of particle physics in the particular setting of broken symmetries described by symmetric spaces. These equations are simple field theory models whose…

The present paper derives systems of partial differential equations that admit a quadratic zero curvature representation for an arbitrary real semisimple Lie algebra. It also determines the general form of Hamilton's principles and…

A rigid body in an ideal fluid is an important example of Hamiltonian systems on a dual to the semidirect product Lie algebra $e(3) = so(3)\ltimes\mathbb R^3$. We present the bi-Hamiltonian structure and the corresponding variables of…

We consider Lagrangians in Hamilton's principle defined on the tangent space $TG$ of a Lie group $G$. Invariance of such a Lagrangian under the action of $G$ leads to the symmetry-reduced Euler-Lagrange equations called the Euler-Poincar\'e…

Hamilton's principle is used to extend for the Toda lattice ODEs to systems of PDEs called the Toda lattice strand equations (T-Strands). The T-Strands in the $n$-particle Toda case comprise $4n-2$ quadratically nonlinear PDEs in one space…

The classical theory of Riemann ellipsoids is formulated naturally as a gauge theory based on a principal G-bundle ${\cal P}$. The structure group G=SO(3) is the vorticity group, and the bundle ${\cal P}=GL_+(3, R})$ is the connected…

A discrete version of Lagrangian reduction is developed in the context of discrete time Lagrangian systems on $G\times G$, where $G$ is a Lie group. We consider the case when the Lagrange function is invariant with respect to the action of…

Dirichlet's problem for the dynamics of fluid bodies with ellipsoidal shape can be formulated as a Hamiltonian system invariant under the action of a symmetry Lie group. I apply methods from Hamiltonian bifurcation theory to the study of…

In this study, we discuss an approximate set of equations describing water wave propagating in deep water. These generalized Klein-Gordon (gKG) equations possess a variational formulation, as well as a canonical Hamiltonian and…

A notion of implicit difference equation on a Lie groupoid is introduced and an algorithm for extracting the integrable part (backward or/and forward) is formulated. As an application, we prove that discrete Lagrangian dynamics on a Lie…

We discuss three examples in which one may extend integrable Euler--Poincar\'e ODEs to integrable Euler--Poincar\'e PDEs in the matrix G-Strand context. After describing matrix G-Strand examples for $SO(3)$ and $SO(4)$ we turn our attention…

We point out, and draw some consequences of, the fact that the Poisson Lie group G* dual to G=GL_n(C) (with its standard complex Poisson structure) may be identified with a certain moduli space of meromorphic connections on the unit disc…

This paper studies the Poisson equation for the $G_2$-Laplacian on 3-forms on the 7-sphere that are invariant under a transitive group action. We establish the existence and uniqueness of $G$-invariant solutions for $G=SU(4),\: Spin(7),\:…

In this paper, discrete analogues of Euler-Poincar\'{e} and Lie-Poisson reduction theory are developed for systems on finite dimensional Lie groups $G$ with Lagrangians $L:TG \to {\mathbb R}$ that are $G$-invariant. These discrete equations…

The semi-classical spectrum of the Homogeneous sine-Gordon theories associated with an arbitrary compact simple Lie group G is obtained and shown to be entirely given by solitons. These theories describe quantum integrable massive…

Recent theoretical work has developed the Hamilton's-principle analog of Lie-Poisson Hamiltonian systems defined on semidirect products. The main theoretical results are twofold: (1) Euler-Poincar\'e equations (the Lagrangian analog of…

For a discrete mechanical system on a Lie group $G$ determined by a (reduced) Lagrangian $\ell$ we define a Poisson structure via the pull-back of the Lie-Poisson structure on the dual of the Lie algebra ${\mathfrak g}^*$ by the…

Dynamical equations in generalized hydrodynamics (GHD), a hydrodynamic theory for integrable quantum systems at the Euler scale, take a rather simple form, even though an infinite number of conserved charges are taken into account. We show…