Anti-Newtonian Dynamics
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Transcript of Anti-Newtonian Dynamics
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Anti-Newtonian Dynamics
J. C. SprottDepartment of Physics
University of Wisconsin – Madison
(in collaboration with Vladimir Zhdankin)
Presented at the
TAAPT Conference
in Martin, Tennessee
on March 27, 2010
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Newton’s Laws of MotionIsaac 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 opposite in direction to the force exerted by object 2 on object 1.
3. If object 1 and object 2 interact, the force exerted by object 1 on object 2 is equal in magnitude and in the same direction as the force exerted by object 2 on object 1.
“Anti-Newtonian”
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Force Direction Newtonian Forces:
Anti-Newtonian Forces:
Rabbit Fox
Earth Moon
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Force Magnitude Gravitational Forces:
Spring Forces:
Etc. …
221
rmmGF
m1 m2
krF
r
krF
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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
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Elastic Collisions (1-D)
Newtonian Forces:
Anti-Newtonian Forces:
0
0
2v
mmm
v
vmmmm
v
rf
fr
rf
rff
mf mr
v0
0
0
2v
mmm
v
vmmmm
v
rf
fr
rf
rff
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Friction
Newton’s Second Law: F = ma = r – bv
Interaction force Friction force
Parameters: Mass: m Force law: Friction: b
m
v
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2-Body Newtonian Dynamics Attractive Forces (eg: gravity):
Repulsive Forces (eg: electric):
+
Bound periodic orbitsor unbounded orbits
Unbounded orbits
No chaos!
+
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3-Body Gravitational Dynamics
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3-Body Eelectrostatic Dynamics
-0.5 < < 0
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1 Fox, 1 Rabbit, 1-D, Periodic
mf = 1mr = 1bf = 1br = 2 = 0
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1 Fox, 1 Rabbit, 2-D, Periodic
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1 Fox, 1 Rabbit, 2-D, Quasiperiodic
mf = 1mr = 2bf = 0br = 0 = -1
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1 Fox, 1 Rabbit, 2-D, Quasiperiodic
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1 Fox, 1 Rabbit, 2-D, Quasiperiodic
mf = 2mr = 1bf = 0.1br = 1 = -1
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1 Fox, 1 Rabbit, 2-D, Quasiperiodic
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1 Fox, 1 Rabbit, 2-D, Chaotic
mf = 1mr = 0.5bf = 1br = 2 = -1
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1 Fox, 1 Rabbit, 2-D, Chaotic
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2 Foxes, 1 Rabbit, 2-D, Chaotic
mf = 2mr = 1bf = 1br = 3 = -1
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2 Foxes, 1 Rabbit, 2-D, Chaotic
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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?
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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)