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```Anti-Newtonian Dynamics
J. C. Sprott
Department of Physics
University of Wisconsin – Madison
(in collaboration with Vladimir Zhdankin)
Presented at the
TAAPT Conference
in Martin, Tennessee
on March 27, 2010
Newton’s Laws of Motion
Isaac Newton, Philosophiæ Naturalis Principia Mathematica (1687)
1. An object moves with a velocity that is constant in magnitude and
direction, unless acted upon by a nonzero net force.
2. The acceleration of an object is directly proportional to the net
force acting on it and inversely proportional to its mass (F = ma).
3. If object 1 and object 2 interact, the force exerted by object 1 on
object 2 is equal in magnitude but
and opposite
in the same
in direction
directiontoasthe
theforce
force
exerted by object 2 on object 1.
“Anti-Newtonian”
Force Direction

Newtonian Forces:
Earth

Moon
Anti-Newtonian Forces:
Rabbit
Fox
Force Magnitude

Gravitational Forces:
r
F G
m1


m1m2
r2
Spring Forces:
Etc. …
F  kr
F  kr
m2
Conservation Laws

Newtonian Forces:
 Kinetic
+ potential energy is
conserved
 Linear momentum is conserved
 Center of mass moves with
constant velocity

Anti-Newtonian Forces:
 Energy
and momentum are not
usually conserved
 Center of mass can accelerate
Elastic Collisions (1-D)
v0
mf
mr
vf 

Newtonian Forces:
vr 
vf 

Anti-Newtonian Forces:
vr 
m f  mr
m f  mr
2m f
m f  mr
m f  mr
m f  mr
2m f
m f  mr
v0
v0
v0
v0
Friction
v
m

Newton’s Second Law:

F = ma = r  – bv
Interaction force Friction force

Parameters:
Mass: m
 Force law: 
 Friction: b

2-Body Newtonian Dynamics

Attractive Forces (eg: gravity):
Bound periodic orbits
or unbounded orbits

Repulsive Forces (eg: electric):
+
+
Unbounded orbits
No chaos!
3-Body Gravitational Dynamics
3-Body Eelectrostatic Dynamics
-0.5 <  < 0
1 Fox, 1 Rabbit, 1-D, Periodic
mf = 1
mr = 1
bf = 1
br = 2
=0
1 Fox, 1 Rabbit, 2-D, Periodic
1 Fox, 1 Rabbit, 2-D, Quasiperiodic
mf = 1
mr = 2
bf = 0
br = 0
 = -1
1 Fox, 1 Rabbit, 2-D, Quasiperiodic
1 Fox, 1 Rabbit, 2-D, Quasiperiodic
mf = 2
mr = 1
bf = 0.1
br = 1
 = -1
1 Fox, 1 Rabbit, 2-D, Quasiperiodic
1 Fox, 1 Rabbit, 2-D, Chaotic
mf = 1
mr = 0.5
bf = 1
br = 2
 = -1
1 Fox, 1 Rabbit, 2-D, Chaotic
2 Foxes, 1 Rabbit, 2-D, Chaotic
mf = 2
mr = 1
bf = 1
br = 3
 = -1
2 Foxes, 1 Rabbit, 2-D, Chaotic
Summary

Richer dynamics than usual case

Chaos with only two bodies in 2-D

Energy and momentum not
conserved

Bizarre collision behavior

More variety (ffr, rrf, …)

Anti-special relativity?

Anti-Bohr atom?
References

http://sprott.physics.wisc.edu/
lectures/antinewt.ppt (this talk)

http://sprott.physics.wisc.edu/pubs/
paper339.htm (written version)

[email protected] (contact
me)
```
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