1,603 research outputs found
Goldfish geodesics and Hamiltonian reduction of matrix dynamics
We relate free vector dynamics to the eigenvalue motion of a time-dependent
real-symmetric NxN matrix, and give a geodesic interpretation to Ruijsenaars
Schneider models.Comment: 8 page
Periodic motions galore: How to modify nonlinear evolution equations so that they feature a lot of periodic solutions
A simple trick is illustrated, whereby nonlinear evolution equations can be
modified so that they feature a lot - or, in some cases, only -- periodic
solutions. Several examples (ODEs and PDEs) are exhibited.Comment: arxiv version is already officia
Bianchi Cosmologies with Anisotropic Matter: Locally Rotationally Symmetric Models
The dynamics of cosmological models with isotropic matter sources (perfect
fluids) is extensively studied in the literature; in comparison, the dynamics
of cosmological models with anisotropic matter sources is not. In this paper we
consider spatially homogeneous locally rotationally symmetric solutions of the
Einstein equations with a large class of anisotropic matter models including
collisionless matter (Vlasov), elastic matter, and magnetic fields. The
dynamics of models of Bianchi types I, II, and IX are completely described; the
two most striking results are the following: (i) There exist matter models,
compatible with the standard energy conditions, such that solutions of Bianchi
type IX (closed cosmologies) need not necessarily recollapse; there is an open
set of forever expanding solutions. (ii) Generic type IX solutions associated
with a matter model like Vlasov matter exhibit oscillatory behavior toward the
initial singularity. This behavior differs significantly from that of
vacuum/perfect fluid cosmologies; hence "matter matters". Finally, we indicate
that our methods can probably be extended to treat a number of open problems,
in particular, the dynamics of Bianchi type VIII and Kantowski-Sachs solutions.Comment: 64 pages, 19 Figure
A discrete linearizability test based on multiscale analysis
In this paper we consider the classification of dispersive linearizable partial difference equations defined on a quad-graph by the multiple scale reduction around their harmonic solution. We show that the A1, A2 and A3 linearizability conditions restrain the number of the parameters which enter into the equation. A subclass of the equations which pass the A3 C-integrability conditions can be linearized by a Möbius transformation
Observable algebra for the rational and trigonometric Euler Calogero Moser models
We construct polynomial Poisson algebras of observables for the classical
Euler-Calogero-Moser (ECM) models. The conserved Hamiltonians and symmetry
algebras derived in a previous work are subsets of these algebras. We define
their linear, limits, realizing \w_{\infty} type
algebras coupled to current algebras.Comment: 11 pages; Latex; PAR LPTHE 94-16 Misprints and minor mistakes
corrected; references update
Knizhnik-Zamolodchikov equations and the Calogero-Sutherland-Moser integrable models with exchange terms
It is shown that from some solutions of generalized Knizhnik-Zamolodchikov
equations one can construct eigenfunctions of the Calogero-Sutherland-Moser
Hamiltonians with exchange terms, which are characterized by any given
permutational symmetry under particle exchange. This generalizes some results
previously derived by Matsuo and Cherednik for the ordinary
Calogero-Sutherland-Moser Hamiltonians.Comment: 13 pages, LaTeX, no figures, to be published in J. Phys.
On frequencies of small oscillations of some dynamical systems associated with root systems
In the paper by F. Calogero and author [Commun. Math. Phys. 59 (1978)
109-116] the formula for frequencies of small oscillations of the Sutherland
system ( case) was found. In present note the generalization of this
formula for the case of arbitrary root system is given.Comment: arxiv version is already officia
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