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)

sprott@physics.wisc.edu (contact

me)