4,400 research outputs found
The Liouville-type theorem for integrable Hamiltonian systems with incomplete flows
For integrable Hamiltonian systems with two degrees of freedom whose
Hamiltonian vector fields have incomplete flows, an analogue of the Liouville
theorem is established. A canonical Liouville fibration is defined by means of
an "exact" 2-parameter family of flat polygons equipped with certain pairing of
sides. For the integrable Hamiltonian systems given by the vector field
on where
is a complex polynomial in 2 variables, geometric properties of
Liouville fibrations are described.Comment: 6 page
Slow Schroedinger dynamics of gauged vortices
Multivortex dynamics in Manton's Schroedinger--Chern--Simons variant of the
Landau-Ginzburg model of thin superconductors is studied within a moduli space
approximation. It is shown that the reduced flow on M_N, the N vortex moduli
space, is hamiltonian with respect to \omega_{L^2}, the L^2 Kaehler form on
\M_N. A purely hamiltonian discussion of the conserved momenta associated with
the euclidean symmetry of the model is given, and it is shown that the
euclidean action on (M_N,\omega_{L^2}) is not hamiltonian. It is argued that
the N=3 flow is integrable in the sense of Liouville. Asymptotic formulae for
\omega_{L^2} and the reduced Hamiltonian for large intervortex separation are
conjectured. Using these, a qualitative analysis of internal 3-vortex dynamics
is given and a spectral stability analysis of certain rotating vortex polygons
is performed. Comparison is made with the dynamics of classical fluid point
vortices and geostrophic vortices.Comment: 22 pages, 2 figure
Hamiltonian evolutions of twisted gons in \RP^n
In this paper we describe a well-chosen discrete moving frame and their
associated invariants along projective polygons in \RP^n, and we use them to
write explicit general expressions for invariant evolutions of projective
-gons. We then use a reduction process inspired by a discrete
Drinfeld-Sokolov reduction to obtain a natural Hamiltonian structure on the
space of projective invariants, and we establish a close relationship between
the projective -gon evolutions and the Hamiltonian evolutions on the
invariants of the flow. We prove that {any} Hamiltonian evolution is induced on
invariants by an evolution of -gons - what we call a projective realization
- and we give the direct connection. Finally, in the planar case we provide
completely integrable evolutions (the Boussinesq lattice related to the lattice
-algebra), their projective realizations and their Hamiltonian pencil. We
generalize both structures to -dimensions and we prove that they are
Poisson. We define explicitly the -dimensional generalization of the planar
evolution (the discretization of the -algebra) and prove that it is
completely integrable, providing also its projective realization
Covariant Lattice Theory and t'Hooft's Formulation
We show that 't Hooft's representation of (2+1)-dimensional gravity in terms
of flat polygonal tiles is closely related to a gauge-fixed version of the
covariant Hamiltonian lattice theory. 't Hooft's gauge is remarkable in that it
leads to a Hamiltonian which is a linear sum of vertex Hamiltonians, each of
which is defined modulo . A cyclic Hamiltonian implies that ``time'' is
quantized. However, it turns out that this Hamiltonian is {\it constrained}. If
one chooses an internal time and solves this constraint for the ``physical
Hamiltonian'', the result is not a cyclic function. Even if one quantizes {\it
a la Dirac}, the ``internal time'' observable does not acquire a discrete
spectrum. We also show that in Euclidean 3-d lattice gravity, ``space'' can be
either discrete or continuous depending on the choice of quantization. Finally,
we propose a generalization of 't Hooft's gauge for Hamiltonian lattice
formulations of topological gravity dimension 4.Comment: 10 pages of text. One figure available from J.A. Zapata upon reques
Canonical Quantization of Gravitating Point Particles in 2+1 Dimensions
A formalism previously introduced by the author using tesselated Cauchy
surfaces is applied to define a quantized version of gravitating point
particles in 2+1 dimensions. We observe that this is the first model whose
quantum version automatically discretizes time. But also spacelike distances
are discretized in a very special way.Comment: 14 pages (TeX), 3 figures (Postscript), Utrecht THU-93/1
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