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Monodromy in the resonant swing spring
This paper shows that an integrable approximation of the spring pendulum,
when tuned to be in resonance, has monodromy. The stepwise precession
angle of the swing plane of the resonant spring pendulum is shown to be a
rotation number of the integrable approximation. Due to the monodromy, this
rotation number is not a globally defined function of the integrals. In fact at
lowest order it is given by where and are functions of the
integrals. The resonant swing spring is therefore a system where monodromy has
easily observed physical consequences.Comment: 30 pages, 5 figure
Developments in Random Matrix Theory
In this preface to the Journal of Physics A, Special Edition on Random Matrix
Theory, we give a review of the main historical developments of random matrix
theory. A short summary of the papers that appear in this special edition is
also given.Comment: 22 pages, Late
On a class of three-dimensional integrable Lagrangians
We characterize non-degenerate Lagrangians of the form such that the corresponding Euler-Lagrange equations are integrable by the method of
hydrodynamic reductions. The integrability conditions constitute an
over-determined system of fourth order PDEs for the Lagrangian density ,
which is in involution and possess interesting differential-geometric
properties. The moduli space of integrable Lagrangians, factorized by the
action of a natural equivalence group, is three-dimensional. Familiar examples
include the dispersionless Kadomtsev-Petviashvili (dKP) and the Boyer-Finley
Lagrangians, and ,
respectively. A complete description of integrable cubic and quartic
Lagrangians is obtained. Up to the equivalence transformations, the list of
integrable cubic Lagrangians reduces to three examples, There exists a
unique integrable quartic Lagrangian, We
conjecture that these examples exhaust the list of integrable polynomial
Lagrangians which are essentially three-dimensional (it was verified that there
exist no polynomial integrable Lagrangians of degree five). We prove that the
Euler-Lagrange equations are integrable by the method of hydrodynamic
reductions if and only if they possess a scalar pseudopotential playing the
role of a dispersionless `Lax pair'. MSC: 35Q58, 37K05, 37K10, 37K25. Keywords:
Multi-dimensional Dispersionless Integrable Systems, Hydrodynamic Reductions,
Pseudopotentials.Comment: 12 pages A4 format, standard Latex 2e. In the file progs.tar we
include the programs needed for computations performed in the paper. Read
1-README first. The new version includes two new section
Universal scaling limits of matrix models, and (p,q) Liouville gravity
We show that near a point where the equilibrium density of eigenvalues of a
matrix model behaves like y ~ x^{p/q}, the correlation functions of a random
matrix, are, to leading order in the appropriate scaling, given by determinants
of the universal (p,q)-minimal models kernels. Those (p,q) kernels are written
in terms of functions solutions of a linear equation of order q, with
polynomial coefficients of degree at most p. For example, near a regular edge y
~ x^{1/2}, the (1,2) kernel is the Airy kernel and we recover the Airy law.
Those kernels are associated to the (p,q) minimal model, i.e. the (p,q)
reduction of the KP hierarchy solution of the string equation. Here we consider
only the 1-matrix model, for which q=2.Comment: pdflatex, 44 pages, 2 figure
The meromorphic non-integrability of the three-body problem
We study the planar three-body problem and prove the absence of a complete
set of complex meromorphic first integrals in a neighborhood of the Lagrangian
solution. We use the Ziglin's method and study the monodromy group of the
corresponding normal variational equations.Comment: 17 pages, submitted to Crelle's Journa
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