874 research outputs found
Another New Solvable Many-Body Model of Goldfish Type
A new solvable many-body problem is identified. It is characterized by
nonlinear Newtonian equations of motion ("acceleration equal force") featuring
one-body and two-body velocity-dependent forces "of goldfish type" which
determine the motion of an arbitrary number of unit-mass point-particles in
a plane. The (generally complex) values at time of the
coordinates of these moving particles are given by the eigenvalues of a
time-dependent matrix explicitly known in terms of the 2N
initial data and . This model comes in two different
variants, one featuring 3 arbitrary coupling constants, the other only 2; for
special values of these parameters all solutions are completely periodic with
the same period independent of the initial data ("isochrony"); for other
special values of these parameters this property holds up to corrections
vanishing exponentially as ("asymptotic isochrony").
Other isochronous variants of these models are also reported. Alternative
formulations, obtained by changing the dependent variables from the zeros
of a monic polynomial of degree to its coefficients, are also
exhibited. Some mathematical findings implied by some of these results - such
as Diophantine properties of the zeros of certain polynomials - are outlined,
but their analysis is postponed to a separate paper
Isochronous solutions of Einstein's equations and their Newtonian limit
It has been recently demonstrated that it is possible to construct
isochronous cosmologies, extending to general relativity a result valid for
non-relativistic Hamiltonian systems. In this paper we review these findings
and we discuss the Newtonian limit of these isochronous spacetimes, showing
that it reproduces the analogous findings in the context of non-relativistic
dynamics.Comment: arXiv admin note: text overlap with arXiv:1406.715
Solvable Nonlinear Evolution PDEs in Multidimensional Space
A class of solvable (systems of) nonlinear evolution PDEs in multidimensional
space is discussed. We focus on a rotation-invariant system of PDEs of
Schr\"odinger type and on a relativistically-invariant system of PDEs of
Klein-Gordon type. Isochronous variants of these evolution PDEs are also
considered.Comment: This is a contribution to the Vadim Kuznetsov Memorial Issue on
Integrable Systems and Related Topics, published in SIGMA (Symmetry,
Integrability and Geometry: Methods and Applications) at
http://www.emis.de/journals/SIGMA
Isochronous Spacetimes
The possibility has been recently demonstrated to manufacture
(nonrelativistic, Hamiltonian) many-body problems which feature an isochronous
time evolution with an arbitrarily assigned period yet mimic with good
approximation, or even exactly, any given many-body problem (within a quite
large class, encompassing most of nonrelativistic physics) over times
which may also be arbitrarily large (but of course such that
). In this paper we review and further explore the possibility to
extend this finding to a general relativity context, so that it becomes
relevant for cosmology.Comment: Submitted to Acta Appl. Mat
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