25,777 research outputs found
Local Gram-Schmidt and Covariant Lyapunov Vectors and Exponents for Three Harmonic Oscillator Problems
We compare the Gram-Schmidt and covariant phase-space-basis-vector
descriptions for three time-reversible harmonic oscillator problems, in two,
three, and four phase-space dimensions respectively. The two-dimensional
problem can be solved analytically. The three-dimensional and four-dimensional
problems studied here are simultaneously chaotic, time-reversible, and
dissipative. Our treatment is intended to be pedagogical, for publication in
Communications in Nonlinear Science and Numerical Computation and for use in an
updated version of our book on Time Reversibility, Computer Simulation, and
Chaos. Comments are very welcome.Comment: 25 pages with 12 figures; New Figures 9-12 based on two billion
timesteps rather than the two hundred million used in Version 1; Electronic
publication in Communications in Nonlinear Science and Numerical Computation
scheduled for 1 July 201
What is liquid? Lyapunov instability reveals symmetry-breaking irreversibilities hidden within Hamilton's many-body equations of motion
Typical Hamiltonian liquids display exponential "Lyapunov instability", also
called "sensitive dependence on initial conditions". Although Hamilton's
equations are thoroughly time-reversible, the forward and backward Lyapunov
instabilities can differ, qualitatively. In numerical work, the expected
forward/backward pairing of Lyapunov exponents is also occasionally violated.
To illustrate, we consider many-body inelastic collisions in two space
dimensions. Two mirror-image colliding crystallites can either bounce, or not,
giving rise to a single liquid drop, or to several smaller droplets, depending
upon the initial kinetic energy and the interparticle forces. The difference
between the forward and backward evolutionary instabilities of these problems
can be correlated with dissipation and with the Second Law of Thermodynamics.
Accordingly, these asymmetric stabilities of Hamilton's equations can provide
an "Arrow of Time". We illustrate these facts for two small crystallites
colliding so as to make a warm liquid. We use a specially-symmetrized form of
Levesque and Verlet's bit-reversible Leapfrog integrator. We analyze
trajectories over millions of collisions with several equally-spaced time
reversals.Comment: 13 pages and 11 figures, prepared for Douglas Henderson's 80th
Birthday Symposium at Brigham Young University in August 2014 revised to
incorporate referee's suggestions as an acknowledgmen
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