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Anti-Newtonian Dynamics

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.

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Anti-Newtonian Dynamics

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  1. 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

  2. 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 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”

  3. Force Direction • Newtonian Forces: • Anti-Newtonian Forces: Earth Moon Rabbit Fox

  4. Force Magnitude r • Gravitational Forces: • Spring Forces: • Etc. … m1 m2

  5. 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

  6. Elastic Collisions (1-D) v0 mf mr • Newtonian Forces: • Anti-Newtonian Forces:

  7. Friction v m • Newton’s Second Law: • F = ma = rg – bv Interaction force Friction force • Parameters: • Mass: m • Force law: g • Friction: b

  8. 2-Body Newtonian Dynamics • Attractive Forces (eg: gravity): • Repulsive Forces (eg: electric): Bound periodic orbits or unbounded orbits + + Unbounded orbits No chaos!

  9. 3-Body Gravitational Dynamics

  10. 3-Body Eelectrostatic Dynamics -0.5 < g < 0

  11. 1 Fox, 1 Rabbit, 1-D, Periodic mf = 1 mr = 1 bf = 1 br = 2 g = 0

  12. 1 Fox, 1 Rabbit, 2-D, Periodic

  13. 1 Fox, 1 Rabbit, 2-D, Quasiperiodic mf = 1 mr = 2 bf = 0 br = 0 g = -1

  14. 1 Fox, 1 Rabbit, 2-D, Quasiperiodic

  15. 1 Fox, 1 Rabbit, 2-D, Quasiperiodic mf = 2 mr = 1 bf = 0.1 br = 1 g = -1

  16. 1 Fox, 1 Rabbit, 2-D, Quasiperiodic

  17. 1 Fox, 1 Rabbit, 2-D, Chaotic mf = 1 mr = 0.5 bf = 1 br = 2 g = -1

  18. 1 Fox, 1 Rabbit, 2-D, Chaotic

  19. 2 Foxes, 1 Rabbit, 2-D, Chaotic mf = 2 mr = 1 bf = 1 br = 3 g = -1

  20. 2 Foxes, 1 Rabbit, 2-D, Chaotic

  21. 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?

  22. 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) References

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