50,659 research outputs found
Reminiscences of East Greenwich
Address delivered before the East Greenwich Business Men’s Association by Henry E. Turner, April 11, 1892.
Henry E. Turner, a member of the East Greenwich Business Men’s Association, recounts the early history of notable events and people of the town of East Greenwich, Rhode Island, during the first half of the 19thcentury.https://digitalcommons.providence.edu/ri_history/1005/thumbnail.jp
Geometric Interpretation of Chaos in Two-Dimensional Hamiltonian Systems
Time-independent Hamiltonian flows are viewed as geodesic flows in a curved
manifold, so that the onset of chaos hinges on properties of the curvature
two-form entering into the Jacobi equation. Attention focuses on ensembles of
orbit segments evolved in 2-D potentials, examining how various orbital
properties correlate with the mean value and dispersion, and k, of the
trace K of the curvature. Unlike most analyses, which have attributed chaos to
negative curvature, this work exploits the fact that geodesics can be chaotic
even if K is everywhere positive, chaos arising as a parameteric instability
triggered by regular variations in K along the orbit. For ensembles of fixed
energy, with both regular and chaotic segments, simple patterns connect the
values of and k for different segments, both with each other and with the
short time Lyapunov exponent X. Often, but not always, there is a near one-to-
one correlation between and k, a plot of these quantities approximating a
simple curve. X varies smoothly along this curve, chaotic segments located
furthest from the regular regions tending systematically to have the largest
X's. For regular orbits, and k also vary smoothly with ``distance'' from
the chaotic phase space regions, as probed, e.g., by the location of the
initial condition on a surface of section. Many of these observed properties
can be understood qualitatively in terms of a one-dimensional Mathieu equation.Comment: 16 pages plus 9 figures, LaTeX, no macros required to appear in
Physical Review
Phase mixing in time-independent Hamiltonian systems
Everything you ever wanted to know about what has come to be known as
``chaotic mixing:'' This paper describes the evolution of localised ensembles
of initial conditions in 2- and 3-D time-independent potentials which admit
both regular and chaotic orbits. The coarse-grained approach towards an
invariant, or near-invariant, distribution was probed by tracking (1) phase
space moments through order 4 and (2) binned reduced distributions f(Z_a,Z_b,t)
for a,b=x,y,z,p_x,p_y,p_z, computed at fixed time intervals. For ``unconfined''
chaotic orbits in 2-D systems not stuck near islands by cantori, the moments
evolve exponentially: Quantities like the dispersion in p_x, which start small
and eventually asymptote towards a larger value, initially grow exponentially
in time at a rate comparable to the largest short time Lyapunov exponent.
Quantities like ||, that can start large but eventually asymptote towards
zero, decrease exponentially. With respect to a discrete L^p norm, reduced
distributions f(t) generated from successive decay exponentially towards a
near-invariant f_{niv}, although a plot of Df(t)=||f(t)-f_{niv}|| can exhibit
considerable structure. Regular ensembles behave very differently, both moments
and Df evolving in a fashion better represented by a power law time dependence.
``Confined'' chaotic orbits, initially stuck near regular islands because of
cantori, exhibit an intermediate behaviour. The behaviour of ensembles evolved
in 3-D potentials is qualitatively similar, except that, in this case, it is
relatively likely to find one direction in configuration space which is ``less
chaotic'' than the other two, so that quantities like L_{ab} depend more
sensitively on which phase space variables one tracks.Comment: 19 pages + 11 Postscript figures, latex, no macros. Monthly Notices
of the Royal Astronomical Society, in pres
Chaos and Chaotic Phase Mixing in Galaxy Evolution and Charged Particle Beams
This paper discusses three new issues that necessarily arise in realistic
attempts to apply nonlinear dynamics to galaxy evolution, namely: (i) the
meaning of chaos in many-body systems, (ii) the time-dependence of the bulk
potential, which can trigger intervals of {\em transient chaos}, and (iii) the
self-consistent nature of any bulk chaos, which is generated by the bodies
themselves, rather than imposed externally. Simulations and theory both suggest
strongly that the physical processes associated with galactic evolution should
also act in nonneutral plasmas and charged particle beams. This in turn
suggests the possibility of testing this physics in real laboratory
experiments, an undertaking currently underway.Comment: 16 pages, including 3 figures: an invited talk at the Athens Workshop
on Galaxies and Chaos, Theory and Observation
Invariant distributions and collisionless equilibria
This paper discusses the possibility of constructing time-independent
solutions to the collisionless Boltzmann equation which depend on quantities
other than global isolating integrals such as energy and angular momentum. The
key point is that, at least in principle, a self-consistent equilibrium can be
constructed from any set of time-independent phase space building blocks which,
when combined, generate the mass distribution associated with an assumed
time-independent potential. This approach provides a way to justify
Schwarzschild's (1979) method for the numerical construction of self-consistent
equilibria with arbitrary time-independent potentials, generalising thereby an
approach developed by Vandervoort (1984) for integrable potentials. As a simple
illustration, Schwarzschild's method is reformulated to allow for a
straightforward computation of equilibria which depend only on one or two
global integrals and no other quantities, as is reasonable, e.g., for modeling
axisymmetric configurations characterised by a nonintegrable potential.Comment: 14 pages, LaTeX, no macro
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