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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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