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
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”
Force Direction Newtonian Forces:
Anti-Newtonian Forces:
Rabbit Fox
Earth Moon
Force Magnitude Gravitational Forces:
Spring Forces:
Etc. …
221
rmmGF
m1 m2
krF
r
krF
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)
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
Friction
Newton’s Second Law: F = ma = r – bv
Interaction force Friction force
Parameters: Mass: m Force law: Friction: b
m
v
2-Body Newtonian Dynamics Attractive Forces (eg: gravity):
Repulsive Forces (eg: electric):
+
Bound periodic orbitsor unbounded orbits
Unbounded orbits
No chaos!
+
3-Body Gravitational Dynamics
3-Body Eelectrostatic Dynamics
-0.5 < < 0
1 Fox, 1 Rabbit, 1-D, Periodic
mf = 1mr = 1bf = 1br = 2 = 0
1 Fox, 1 Rabbit, 2-D, Periodic
1 Fox, 1 Rabbit, 2-D, Quasiperiodic
mf = 1mr = 2bf = 0br = 0 = -1
1 Fox, 1 Rabbit, 2-D, Quasiperiodic
1 Fox, 1 Rabbit, 2-D, Quasiperiodic
mf = 2mr = 1bf = 0.1br = 1 = -1
1 Fox, 1 Rabbit, 2-D, Quasiperiodic
1 Fox, 1 Rabbit, 2-D, Chaotic
mf = 1mr = 0.5bf = 1br = 2 = -1
1 Fox, 1 Rabbit, 2-D, Chaotic
2 Foxes, 1 Rabbit, 2-D, Chaotic
mf = 2mr = 1bf = 1br = 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)
