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

Inspiration October 7, 2008 Molecular mechanisms of synaptic growth: insights from the Drosophila neuromuscular junction Kate O'Connor-Giles, UW Department of Genetics

Newton’s Laws of Motion Isaac Newton, Philosophiæ Naturalis Principia Mathematica (1687)Philosophiæ Naturalis Principia Mathematica 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 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 n Newtonian Forces: n Anti-Newtonian Forces: RabbitFox EarthMoon

Force Magnitude n Gravitational Forces: n Spring Forces: n Etc. … m1m1 m2m2 r

Conservation Laws n Newtonian Forces: u Kinetic + potential energy is conserved u Linear momentum is conserved u Center of mass moves with constant velocity n Anti-Newtonian Forces: u Energy and momentum are not usually conserved u Center of mass can accelerate

Elastic Collisions (1-D) n Newtonian Forces: n Anti-Newtonian Forces: mfmf mrmr v0v0

Friction n Newton’s Second Law:  F = ma = r  – bv Interaction force Friction force n Parameters:  Mass: m  Force law:   Friction: b m v

2-Body Newtonian Dynamics n Attractive Forces (eg: gravity): n Repulsive Forces (eg: electric): + Bound periodic orbits or unbounded orbits Unbounded orbits No chaos! +

3-Body Gravitational Dynamics

3-Body Eelectrostatic Dynamics -0.5 <  < 0

1 Fox, 1 Rabbit, 1-D, Periodic m f = 1 m r = 1 b f = 1 b r = 2  = 0

1 Fox, 1 Rabbit, 2-D, Periodic

1 Fox, 1 Rabbit, 2-D, Quasiperiodic m f = 1 m r = 2 b f = 0 b r = 0  = -1

1 Fox, 1 Rabbit, 2-D, Quasiperiodic

m f = 2 m r = 1 b f = 0.1 b r = 1  = -1

1 Fox, 1 Rabbit, 2-D, Quasiperiodic

1 Fox, 1 Rabbit, 2-D, Chaotic m f = 1 m r = 0.5 b f = 1 b r = 2  = -1

1 Fox, 1 Rabbit, 2-D, Chaotic

2 Foxes, 1 Rabbit, 2-D, Chaotic m f = 2 m r = 1 b f = 1 b r = 3  = -1

2 Foxes, 1 Rabbit, 2-D, Chaotic

Review – Am. J. Phys. This delightful paper should be published in AJP. It is a great example of “what if” thinking. What if I make a small change in a well-known physical model: what would be the consequences? In this case, Professor Sprott investigates the consequences of altering Newton’s 3 rd Law so that forces between two particles are equal in both magnitude and direction instead of equal and opposite. There are two great advantages to asking this type of “what if” question. The first is pedagogical – it allows a student to examine a theory from a new point of view and to recognize the particular consequences of an aspect of the theory by deriving the consequences of changing it, which can be a great learning experience. Second, this type of thinking sometimes leads to quite useful new models and even profound breakthroughs. What if the time between two events is not invariant in a transformation between inertial reference frames? That question lead to a profound revision of Newtonian dynamics. Obviously, Professor Sprott’s anti-Newtonian dynamics are not as profound as special relativity, but he nonetheless does an excellent job of coming up with an interesting example of a dynamical system to which the new model might apply – foxes chasing rabbits. I believe Feynman somewhere talks about the value of a “what if” exercise, but unfortunately I can’t remember where.

Summary n Richer dynamics than usual case n Chaos with only two bodies in 2-D n Energy and momentum not conserved n Bizarre collision behavior n More variety (ffr, rrf, …) n Anti-special relativity? n Anti-Bohr atom?

To be continued … October 6, 2009 Simulation of swarming behavior using anti-Newtonian forces Vladimir Zhdankin, UW Department of Physics The emergent behavior of swarming is investigated by using computer simulation. Each biological agent can be represented as a particle being influenced by forces due to the other agents in the system. A short-range repulsive force and long-range attractive force results in cohesive swarming behavior. However, more complicated dynamics can occur when two distinct species are defined to interact with different force laws. In order to recreate predator and prey swarming behavior that has been observed in nature, an "anti- Newtonian" force will be used between the two species, which violates Newton's Third Law. The resulting dynamics display a lush variety of features, including chaos and emergent behavior. The interesting cases will be demonstrated visually through animations that show the simulations unfold.animations

References n lectures/antinewt.ppt (this talk) lectures/antinewt.ppt n paper339.htm (written version) paper339.htm n (contact me)