38 research outputs found
Matrix polynomials, generalized Jacobians, and graphical zonotopes
A matrix polynomial is a polynomial in a complex variable with
coefficients in complex matrices. The spectral curve of a matrix
polynomial is the curve . The set of matrix
polynomials with a given spectral curve is known to be closely related to
the Jacobian of , provided that is smooth. We extend this result to the
case when is an arbitrary nodal, possibly reducible, curve. In the latter
case the set of matrix polynomials with spectral curve turns out to be
naturally stratified into smooth pieces, each one being an open subset in a
certain generalized Jacobian. We give a description of this stratification in
terms of purely combinatorial data and describe the adjacency of strata. We
also make a conjecture on the relation between completely reducible matrix
polynomials and the canonical compactified Jacobian defined by V.Alexeev.Comment: 19 pages, 7 figure
Algebraic geometry and stability for integrable systems
In 1970s, a method was developed for integration of nonlinear equations by
means of algebraic geometry. Starting from a Lax representation with spectral
parameter, the algebro-geometric method allows to solve the system explicitly
in terms of theta functions of Riemann surfaces. However, the explicit formulas
obtained in this way fail to answer qualitative questions such as whether a
given singular solution is stable or not. In the present paper, the problem of
stability for equilibrium points is considered, and it is shown that this
problem can also be approached by means of algebraic geometry
Stability of relative equilibria of multidimensional rigid body
It is a classical result of Euler that the rotation of a torque-free
three-dimensional rigid body about the short or the long axis is stable,
whereas the rotation about the middle axis is unstable. This result is
generalized to the case of a multidimensional body
Curvature of Poisson pencils in dimension three
A Poisson pencil is called flat if all brackets of the pencil can be
simultaneously locally brought to a constant form. Given a Poisson pencil on a
3-manifold, we study under which conditions it is flat. Since the works of
Gelfand and Zakharevich, it is known that a pencil is flat if and only if the
associated Veronese web is trivial. We suggest a simpler obstruction to
flatness, which we call the curvature form of a Poisson pencil. This form can
be defined in two ways: either via the Blaschke curvature form of the
associated web, or via the Ricci tensor of a connection compatible with the
pencil. We show that the curvature form of a Poisson pencil can be given by a
simple explicit formula. This allows us to study flatness of linear pencils on
three-dimensional Lie algebras, in particular those related to the argument
translation method. Many of them appear to be non-flat.Comment: 14 pages, 1 figur
Stability in bi-Hamiltonian systems and multidimensional rigid body
The presence of two compatible Hamiltonian structures is known to be one of
the main, and the most natural, mechanisms of integrability. For every pair of
Hamiltonian structures, there are associated conservation laws (first
integrals). Another approach is to consider the second Hamiltonian structure on
its own as a tensor conservation law. The latter is more intrinsic as compared
to scalar conservation laws derived from it and, as a rule, it is "simpler".
Thus it is natural to ask: can the dynamics of a bi-Hamiltonian system be
understood by studying its Hamiltonian pair, without studying the associated
first integrals?\par In this paper, the problem of stability of equilibria in
bi-Hamiltonian systems is considered and it is shown that the conditions for
nonlinear stability can be expressed in algebraic terms of linearization of the
underlying Poisson pencil. This is used to study stability of stationary
rotations of a free multidimensional rigid body.Comment: Journal of Geometry and Physics, 201
Generalized argument shift method and complete commutative subalgebras in polynomial Poisson algebras
The Mischenko-Fomenko argument shift method allows to construct commutative
subalgebras in the symmetric algebra of a finite-dimensional
Lie algebra . For a wide class of Lie algebras, these commutative
subalgebras appear to be complete, i.e. they have maximal transcendence degree.
However, for many algebras, Mischenko-Fomenko subalgebras are incomplete or
even empty. In this case, we suggest a natural way how to extend
Mischenko-Fomenko subalgebras, and give a completeness criterion for these
extended subalgebras
Flat bi-Hamiltonian structures and invariant densities
A bi-Hamiltonian structure is a pair of Poisson structures ,
which are compatible, meaning that any linear combination is again a Poisson structure. A bi-Hamiltonian
structure is called flat if and
can be simultaneously brought to a constant form in a neighborhood
of a generic point. We prove that a generic bi-Hamiltonian structure on an odd-dimensional manifold is flat if and only if there
exists a local density which is preserved by all vector fields Hamiltonian with
respect to , as well as by all vector fields Hamiltonian with
respect to .Comment: 10 pages; Lett Math Phys (2016
Singularities of integrable systems and nodal curves
The relation between integrable systems and algebraic geometry is known since
the XIXth century. The modern approach is to represent an integrable system as
a Lax equation with spectral parameter. In this approach, the integrals of the
system turn out to be the coefficients of the characteristic polynomial
of the Lax matrix, and the solutions are expressed in terms of theta functions
related to the curve . The aim of the present paper is to show that
the possibility to write an integrable system in the Lax form, as well as the
algebro-geometric technique related to this possibility, may also be applied to
study qualitative features of the system, in particular its singularities.Comment: 30 pages, 1 figure, 2 tables. Partially published as Singularities of
integrable systems and algebraic curves, International Mathematics Research
Notices, doi:10.1093/imrn/rnw168 (2016
A note on relative equilibria of multidimensional rigid body
It is well known that a rotation of a free generic three-dimensional rigid
body is stationary if and only if it is a rotation around one of three
principal axes of inertia. As it was noted by many authors, the analogous
result is true for a multidimensional body: a rotation is stationary if and
only if it is a rotation in the principal axes of inertia, provided that the
eigenvalues of the angular velocity matrix are pairwise distinct. However, if
some eigenvalues of the angular velocity matrix of a stationary rotation
coincide, then it is possible that this rotation has a different nature. A
description of such rotations is given in the present paper
Characterization of steady solutions to the 2D Euler equation
Steady fluid flows have very special topology. In this paper we describe
necessary and sufficient conditions on the vorticity function of a 2D ideal
flow on a surface with or without boundary, for which there exists a steady
flow among isovorticed fields. For this we introduce the notion of an
antiderivative (or circulation function) on a measured graph, the Reeb graph
associated to the vorticity function on the surface, while the criterion is
related to the total negativity of this antiderivative. It turns out that given
topology of the vorticity function, the set of coadjoint orbits of the
symplectomorphism group admitting steady flows with this topology forms a
convex polytope. As a byproduct of the proposed construction, we also describe
a complete list of Casimirs for the 2D Euler hydrodynamics: we define
generalized enstrophies which, along with circulations, form a complete set of
invariants for coadjoint orbits of area-preserving diffeomorphisms on a
surface.Comment: 34 pages, 13 figure