268 research outputs found
Unfreezing Casimir invariants: singular perturbations giving rise to forbidden instabilities
The infinite-dimensional mechanics of fluids and plasmas can be formulated as
"noncanonical" Hamiltonian systems on a phase space of Eulerian variables.
Singularities of the Poisson bracket operator produce singular Casimir elements
that foliate the phase space, imposing topological constraints on the dynamics.
Here we proffer a physical interpretation of Casimir elements as
\emph{adiabatic invariants} ---upon coarse graining microscopic angle
variables, we obtain a macroscopic hierarchy on which the separated action
variables become adiabatic invariants. On reflection, a Casimir element may be
\emph{unfrozen} by recovering a corresponding angle variable; such an increase
in the number of degrees of freedom is, then, formulated as a \emph{singular
perturbation}. As an example, we propose a canonization of the
resonant-singularity of the Poisson bracket operator of the linearized
magnetohydrodynamics equations, by which the ideal obstacle (resonant Casimir
element) constraining the dynamics is unfrozen, giving rise to a tearing-mode
instability
Existence and stability of Camm type steady states in galactic dynamics
We prove the existence and nonlinear stability of Camm type steady states of
the Vlasov-Poisson system in the gravitational case. The paper demonstrates the
effectiveness of an approach to the existence and stability problem for steady
states, which was used in previous work by the authors: The steady states are
obtained as minimizers of an energy-Casimir functional, and from this fact
their dynamical stability is deduced.Comment: 21 pages, LaTe
Flat steady states in stellar dynamics - existence and stability
We consider a special case of the three dimensional Vlasov-Poisson system
where the particles are restricted to a plane, a situation that is used in
astrophysics to model extremely flattened galaxies. We prove the existence of
steady states of this system. They are obtained as minimizers of an
energy-Casimir functional from which fact a certain dynamical stability
property is deduced. From a mathematics point of view these steady states
provide examples of partially singular solutions of the three dimensional
Vlasov-Poisson system.Comment: 25 pages, LaTe
Stability of spherically symmetric steady states in galactic dynamics against general perturbations
Certain steady states of the Vlasov-Poisson system can be characterized as
minimizers of an energy-Casimir functional, and this fact implies a nonlinear
stability property of such steady states. In previous investigations by Y. Guo
and the author stability was obtained only with respect to spherically
symmetric perturbations. In the present investigation we show how to remove
this unphysical restriction.Comment: 19 pages LaTe
Isotropic steady states in galactic dynamics revised
The present paper completes our earlier results on nonlinear stability of
stationary solutions of the Vlasov-Poisson system in the stellar dynamics case.
By minimizing the energy under a mass-Casimir constraint we construct a large
class of isotropic, spherically symmetric steady states and prove their
nonlinear stability against general, i. e., not necessarily symmetric
perturbations. The class is optimal in a certain sense, in particular, it
includes all polytropes of finite mass with decreasing dependence on the
particle energy.Comment: 31 pages, LaTe
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