Curve Flows in Lagrange-Finsler Geometry, Bi-Hamiltonian Structures and Solitons

Details Curve Flows in Lagrange-Finsler Geometry, Bi-Hamiltonian Structures and Solitons Curve Flows in Lagrange-Finsler Geometry, Bi-Hamiltonian Structures and Solitons

2006-09-25

arxiv.org/abs/math-ph/0609070v3

J.Geom.Phys.59:79-103,2009

Physics

Mathematical Physics

latex 2e 50 pages, the manuscript is a Lagrange-Finsler generalization of the solitonic Riemannian formalism from math-ph/0608

Scientific paper

10.1016/j.geomphys.2008.10.006

Methods in Riemann-Finsler geometry are applied to investigate bi-Hamiltonian structures and related mKdV hierarchies of soliton equations derived geometrically from regular Lagrangians and flows of non-stretching curves in tangent bundles. The total space geometry and nonholonomic flows of curves are defined by Lagrangian semisprays inducing canonical nonlinear connections (N-connections), Sasaki type metrics and linear connections. The simplest examples of such geometries are given by tangent bundles on Riemannian symmetric spaces $G/SO(n)$ provided with an N-connection structure and an adapted metric, for which we elaborate a complete classification, and by generalized Lagrange spaces with constant Hessian. In this approach, bi-Hamiltonian structures are derived for geometric mechanical models and (pseudo) Riemannian metrics in gravity. The results yield horizontal/ vertical pairs of vector sine-Gordon equations and vector mKdV equations, with the corresponding geometric curve flows in the hierarchies described in an explicit form by nonholonomic wave maps and mKdV analogs of nonholonomic Schrodinger maps on a tangent bundle.

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