Completely Integrable Gradient Flows
نویسندگان
چکیده
In this paper we exhibit the Toda lattice equations in a double bracket form which shows they are gradient flow equations (on their isospectral set) on an adjoint orbit of a compact Lie group. Representations for the flows are given and a convexity result associated with a momentum map is proved. Some general properties of the double bracket equations are demonstrated, including a discussion of their invariant subspaces, and their function as a Lie algebraic sorter.
منابع مشابه
Gradient flows within plane fields
We consider the dynamics of vector fields on three-manifolds which are constrained to lie within a plane field, such as occurs in nonholonomic dynamics. On compact manifolds, such vector fields force dynamics beyond that of a gradient flow, except in cases where the underlying manifold is topologically simple (i.e., a graphmanifold). Furthermore, there are strong restrictions on the types of gr...
متن کاملToric Integrable Geodesic Flows
By studying completely integrable torus actions on contact manifolds we prove a conjecture of Toth and Zelditch that toric integrable geodesic flows on tori must have flat metrics.
متن کاملOn the Integrability of Geodesic Flows of Submersion Metrics
Suppose we are given a compact Riemannian manifold (Q, g) with a completely integrable geodesic flow. Let G be a compact connected Lie group acting freely on Q by isometries. The natural question arises: will the geodesic flow on Q/G equipped with the submersion metric be integrable? Under one natural assumption, we prove that the answer is affirmative. New examples of manifolds with completely...
متن کاملHessian Riemannian Gradient Flows in Convex Programming
In view of solving theoretically constrained minimization problems, we investigate the properties of the gradient flows with respect to Hessian Riemannian metrics induced by Legendre functions. The first result characterizes Hessian Riemannian structures on convex sets as metrics that have a specific integration property with respect to variational inequalities, giving a new motivation for the ...
متن کامل3-manifolds Admitting Toric Integrable Geodesic Flows
A toric integrable geodesic flow on a manifold M is a completely integrable geodesic flow such that the integrals generate a homogeneous torus action on the punctured cotangent bundle T ∗M \ M . A toric integrable manifold is a manifold which has a toric integrable geodesic flow. Toric integrable manifolds can be characterized as those whose cosphere bundle (the sphere bundle in the cotangent b...
متن کامل