2,925 research outputs found
Smooth Hamiltonian systems with soft impacts
In a Hamiltonian system with impacts (or "billiard with potential"), a point
particle moves about the interior of a bounded domain according to a background
potential, and undergoes elastic collisions at the boundaries. When the
background potential is identically zero, this is the hard-wall billiard model.
Previous results on smooth billiard models (where the hard-wall boundary is
replaced by a steep smooth billiard-like potential) have clarified how the
approximation of a smooth billiard with a hard-wall billiard may be utilized
rigorously. These results are extended here to models with smooth background
potential satisfying some natural conditions. This generalization is then
applied to geometric models of collinear triatomic chemical reactions (the
models are far from integrable -degree of freedom systems with ).
The application demonstrates that the simpler analytical calculations for the
hard-wall system may be used to obtain qualitative information with regard to
the solution structure of the smooth system and to quantitatively assist in
finding solutions of the soft impact system by continuation methods. In
particular, stable periodic triatomic configurations are easily located for the
smooth highly-nonlinear two and three degree of freedom geometric models.Comment: 33 pages, 8 figure
Approximating multi-dimensional Hamiltonian flows by billiards
Consider a family of smooth potentials , which, in the limit
, become a singular hard-wall potential of a multi-dimensional
billiard. We define auxiliary billiard domains that asymptote, as
to the original billiard, and provide asymptotic expansion of
the smooth Hamiltonian solution in terms of these billiard approximations. The
asymptotic expansion includes error estimates in the norm and an
iteration scheme for improving this approximation. Applying this theory to
smooth potentials which limit to the multi-dimensional close to ellipsoidal
billiards, we predict when the separatrix splitting persists for various types
of potentials
Synchronizabilities of Networks: A New index
The random matrix theory is used to bridge the network structures and the
dynamical processes defined on them. We propose a possible dynamical mechanism
for the enhancement effect of network structures on synchronization processes,
based upon which a dynamic-based index of the synchronizability is introduced
in the present paper.Comment: 4pages, 2figure
From low-momentum interactions to nuclear structure
We present an overview of low-momentum two-nucleon and many-body interactions
and their use in calculations of nuclei and infinite matter. The softening of
phenomenological and effective field theory (EFT) potentials by renormalization
group (RG) transformations that decouple low and high momenta leads to greatly
enhanced convergence in few- and many-body systems while maintaining a
decreasing hierarchy of many-body forces. This review surveys the RG-based
technology and results, discusses the connections to chiral EFT, and clarifies
various misconceptions.Comment: 76 pages, 57 figures, two figures updated, published versio
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