2,219 research outputs found
A characterization of the Lie Algebra Rank Condition by transverse periodic Functions
The Lie Algebra Rank Condition (LARC) plays a central role in nonlinear systems control theory. The present paper establishes that the satisfaction of this condition by a set of smooth control vector fields is equivalent to the existence of smooth transverse periodic functions. The proof here enclosed is constructive and provides an explicit method for the synthesis of such functions
Poisson brackets with prescribed Casimirs
We consider the problem of constructing Poisson brackets on smooth manifolds
with prescribed Casimir functions. If is of even dimension, we achieve
our construction by considering a suitable almost symplectic structure on ,
while, in the case where is of odd dimension, our objective is achieved by
using a convenient almost cosymplectic structure. Several examples and
applications are presented.Comment: 24 page
Integrable Floquet dynamics
We discuss several classes of integrable Floquet systems, i.e. systems which
do not exhibit chaotic behavior even under a time dependent perturbation. The
first class is associated with finite-dimensional Lie groups and
infinite-dimensional generalization thereof. The second class is related to the
row transfer matrices of the 2D statistical mechanics models. The third class
of models, called here "boost models", is constructed as a periodic interchange
of two Hamiltonians - one is the integrable lattice model Hamiltonian, while
the second is the boost operator. The latter for known cases coincides with the
entanglement Hamiltonian and is closely related to the corner transfer matrix
of the corresponding 2D statistical models. We present several explicit
examples. As an interesting application of the boost models we discuss a
possibility of generating periodically oscillating states with the period
different from that of the driving field. In particular, one can realize an
oscillating state by performing a static quench to a boost operator. We term
this state a "Quantum Boost Clock". All analyzed setups can be readily realized
experimentally, for example in cod atoms.Comment: 18 pages, 2 figures; revised version. Submission to SciPos
Global Action-Angle Variables for Non-Commutative Integrable Systems
In this paper we analyze the obstructions to the existence of global
action-angle variables for regular non-commutative integrable systems (NCI
systems) on Poisson manifolds. In contrast with local action-angle variables,
which exist as soon as the fibers of the momentum map of such an integrable
system are compact, global action-angle variables rarely exist. This fact was
first observed and analyzed by Duistermaat in the case of Liouville integrable
systems on symplectic manifolds and later by Dazord-Delzant in the case of
non-commutative integrable systems on symplectic manifolds. In our more general
case where phase space is an arbitrary Poisson manifold, there are more
obstructions, as we will show both abstractly and on concrete examples. Our
approach makes use of a few new features which we introduce: the action bundle
and the action lattice bundle of the NCI system (these bundles are canonically
defined) and three foliations (the action, angle and transverse foliation),
whose existence is also subject to obstructions, often of a cohomological
nature
Singularities of bi-Hamiltonian systems
We study the relationship between singularities of bi-Hamiltonian systems and
algebraic properties of compatible Poisson brackets. As the main tool, we
introduce the notion of linearization of a Poisson pencil. From the algebraic
viewpoint, a linearized Poisson pencil can be understood as a Lie algebra with
a fixed 2-cocycle. In terms of such linearizations, we give a criterion for
non-degeneracy of singular points of bi-Hamiltonian systems and describe their
types
Open problems, questions, and challenges in finite-dimensional integrable systems
The paper surveys open problems and questions related to different aspects
of integrable systems with finitely many degrees of freedom. Many of the open
problems were suggested by the participants of the conference “Finite-dimensional
Integrable Systems, FDIS 2017” held at CRM, Barcelona in July 2017.Postprint (updated version
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