Anti-Newtonian Dynamics Animations
Vladimir Zhdankin and J. C. Sprott
Department of Physics, University of Wisconsin,
Madison,
WI 53706, USA
April 28, 2009
This animation shows a trivial example of a
fox
encircling a rabbit with both in synchronous concentric circular orbits
using the anti-Newtonian approximation (forces are
equal in magnitude and
direction). The fox has half the mass of the rabbit, and the motion is
frictionless The force
law is
1/R in two spatial dimensions. (Actually, the force law doesn't matter
for this case since the separation is constant.) The
dynamics are periodic with an invariant circle in the
8-dimensional
phase space.
This animation shows an example of a
fox
encircling a rabbit with both in off-centered nested elliptical orbits
using the anti-Newtonian approximation (forces are
equal in magnitude and
direction). The fox has half the mass of the rabbit, and the motion is
frictionless The force
law is
1/R2 in two spatial dimensions. The
dynamics are periodic with an invariant circle in the
8-dimensional
phase space.
This animation shows the motion of a fox
chasing a rabbit using the anti-Newtonian approximation (forces are
equal in magnitude and
direction). The fox has half the mass of the rabbit, and the motion is
frictionless The force
law is
1/R in two spatial dimensions. The
dynamics are quasi-periodic with an invariant torus in the
8-dimensional
phase space.
This animation shows the motion of a fox
chasing a rabbit using the anti-Newtonian approximation (forces are
equal in magnitude and
direction). The fox has
twice the mass of the rabbit and 1/10 of the friction. The force law is
1/R in two spatial dimensions. The
dynamics are quasi-periodic with an attracting torus in the
8-dimensional
phase space. The distance of closest approach is 1.637298.
This animation shows the motion of a fox
chasing a rabbit using the anti-Newtonian approximation (forces are
equal in magnitude and
direction). The fox has
twice the mass of the rabbit and half the friction. The force law is
1/R in two spatial dimensions. The
dynamics are chaotic with a strange attractor in the 8-dimensional
phase space. The distance of closest approach is ~0.1.
This animation shows the motion of two foxes
chasing a rabbit using the anti-Newtonian approximation (forces are
equal in magnitude and
direction). The foxes have
twice the mass of the rabbit and 1/3 of the friction. The force law is
1/R in two spatial dimensions, and the foxes do not interact. The
dynamics are chaotic with a largest
Lyapunov exponent of 0.1346 and a strange attractor in the
12-dimensional phase space.
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