2 research outputs found
Gradient catastrophe and flutter in vortex filament dynamics
Gradient catastrophe and flutter instability in the motion of vortex filament
within the localized induction approximation are analyzed. It is shown that the
origin if this phenomenon is in the gradient catastrophe for the dispersionless
Da Rios system which describes motion of filament with slow varying curvature
and torsion. Geometrically this catastrophe manifests as a rapid oscillation of
a filament curve in a point that resembles the flutter of airfoils.
Analytically it is the elliptic umbilic singularity in the terminology of the
catastrophe theory. It is demonstrated that its double scaling regularization
is governed by the Painlev\'e-I equation.Comment: 11 pages, 3 figures, typos corrected, references adde
Generalized St\"ackel Transform and Reciprocal Transformations for Finite-Dimensional Integrable Systems
We present a multiparameter generalization of the St\"ackel transform (the
latter is also known as the coupling-constant metamorphosis) and show that
under certain conditions this generalized St\"ackel transform preserves the
Liouville integrability, noncommutative integrability and superintegrability.
The corresponding transformation for the equations of motion proves to be
nothing but a reciprocal transformation of a special form, and we investigate
the properties of this reciprocal transformation.
Finally, we show that the Hamiltonians of the systems possessing separation
curves of apparently very different form can be related through a suitably
chosen generalized St\"ackel transform.Comment: 21 pages, LaTeX 2e, no figures; major revision; Propositions 2 and 7
and several new references adde