32 research outputs found
Singular Poisson reduction of cotangent bundles
We consider the Poisson reduced space with respect to a cotangent
lifted action. It is assumed that is a compact Lie group which acts by
isometries on the Riemannian manifold and that the action on is of
single isotropy type. Realizing as a Weinstein space we determine
the induced Poisson structure and its symplectic leaves. We thus extend the
Weinstein construction for principal fiber bundles to the case of surjective
Riemannian submersions .Comment: 28 page
Probabilistic representation of helicity in viscous fluids
It is shown that the helicity of three dimensional viscous incompressible
flow can be identified with the overall linking of the fluid's initial
vorticity to the expectation of a stochastic mean field limit. The relevant
mean field limit is obtained by following the Lagrangian paths in the
stochastic Hamiltonian interacting particle system of [S. Hochgerner, Proc. R.
Soc. A 474:20180178].Comment: v2: fixed a mistake in the setup; v3: streamlined exposition
Nonlinear feedback, double bracket dissipation and port control of Lie-Poisson systems
Methods from controlled Lagrangians, double bracket dissipation and
interconnection and damping assignment -- passivity based control (IDA-PBC) are
used to construct nonlinear feedback controls which (asymptotically) stabilize
previously unstable equilibria of Lie-Poisson Hamiltonian systems. The results
are applied to find an asymptotically stabilizing control for the rotor driven
satellite, and a stabilizing control for Hall magnetohydrodynamic flow
Chaplygin systems associated to Cartan decompositions of semi-simple Lie groups
AbstractWe relate a Chaplygin type system to a Cartan decomposition of a real semi-simple Lie group. The resulting system is described in terms of the structure theory associated to the Cartan decomposition. It is shown to possess a preserved measure and when internal symmetries are present these are factored out via a process called truncation. Furthermore, a criterion for Hamiltonizability of the system on the so-called ultimate reduced level is given. As important special cases we find the Chaplygin ball rolling on a table and the rubber ball rolling over another ball