1. Slide 1/25 Amplitude Quantization as a Fundamental Property of Coupled Oscillator Systems W. J. Wilson Department of Engineering and Physics University of Central Oklahoma Edmond, OK 73034 email: wwilson@uco.edu

2. Slide 2/25 Outline I.Introduction II.Argumental Oscillator (Doubochinski Pendulum) III.“Theory” of Amplitude Quantization IV.Oscillator Trap V.Self-organization Behavior VI.Implications and Conclusions

3. Slide 3/25 Quantum Trap IT’S A TRAP!

4. Slide 4/25

5. Slide 5/25 Argumentally Coupled Oscillators Introduced by Russian physicists to describe classical systems where the configuration of an oscillating system, enters as a variable into the functional expression for the external, oscillating force acting upon it The possibility of self-regulation of energy-exchange is a general characteristic of argumental oscillations.

6. Slide 6/25 Classical Problems 1.Concept of force implies a rigid, “slave-like” obeisance of a system to an external “applied force.” 2.A “force” can act, without itself being changed or being influenced by the system upon which it is acting. Newton’s third law of action and reaction is not enough to remedy that flaw, because it assumes a simplistic form of point-to-point vector action. 3.Attempt to break up the interactions of physical systems into a sum of supposedly elementary, point-to- point actions.

7. Slide 7/25 Classical Coupled Oscillators The idea of an external force, while it may serve as a “useful fiction” for the treatment of certain problems in mechanics, should never be taken as more than that. An “external force” is a simplistic approximation, for an interaction of physical systems Interacting systems never exist as isolated entities in the first place, but only as subsystems of the Universe as a whole, as an organic totality.

8. Slide 8/25 Doubochinski Pendulum

9. Slide 9/25 Doubochinski Pendulum Low Friction Pivot Pendulum with iron mass (f 0 = 1-2 Hz) Alternating Magnetic Field at base (f = 20 – 3000 Hz) driven by V = V 0 sin (2π f t)

10. Slide 10/25 Small Amplitude Oscillations Give familiar resonance physics for Zone 1 oscillations More interesting to look at nonlinear effects and f ≈ 10f 0 -1000f 0

11. Slide 11/25 Yields Quantized Amplitudes f = 50 Hz, f 0 = 2 Hz Stable amplitudes are quantized System “Choice” of stable mode determined by i.c.’s Remarkably stable, large disturbances can cause the pendulum to “jump” from one stable mode to another

12. Slide 12/25 Period for all Oscillations, ~T 0

13. Slide 13/25 Energy Quantized Like Harmonic Oscillator E = E 0 (n + ½)

14. Slide 14/25 Computational Analysis Numerical integration is surprisingly ineffective.

15. Slide 15/25 Perturbative Schemes More Effective In this case, since the total number of decelerating half-cycles will be one less than the number of accelerating half-cycles, after cancellation of pairs of oppositely acting half-cycles, the net effect will be equivalent to that of the first half cycle. In this case, the pendulum will gain energy. But require assuming oscillates with ~T 0

16. Slide 16/25 Phase Dependence lChanges in the pendulum’s velocity, and also in the time during which the pendulum remains in the interaction zone, as a result of the interaction with the electromagnet. lA surprising asymmetry arises in the process, leading to a situation, in which the pendulum can draw a net positive power from the magnet, even without a tight correlation of phase having been established.

17. Slide 17/25 Ratio f/f0101103105107109111 Observed Amplitude30º43º53º60º68º74º Calculated Amplitude23º39º50º59º66º72º

18. Slide 18/25 Multiple pendulums with different natural frequencies can be driven by a single high-frequency magnetic field

19. Slide 19/25 Trap Oscillator

20. Slide 20/25 Spatial Analogue Point-like absorber Effective Size

21. Slide 21/25 Gravitational Segregation Agitate, f

22. Slide 22/25 Possible Applications Electric motors having a discrete multiplicity of rotor speeds for one and the same frequency of the supplied current Vibrational Methods for Sorting Cooling Processes

23. Slide 23/25 Conclusions Argumental oscillations can efficiently couple oscillation processes at frequencies differing by two or more orders of magnitude This coupling can be used to transfer energy into or out of trapped oscillators Fundamental physics can be investigated using particle traps and their interactions with oscillatory fields at much higher frequencies. Paradoxically one can energize to cool, transmit to receive, and add kinetic energy to reach lower energy state.

24. Slide 24/25

25. Slide 25/25 References J. Tennebaum, “Amplitude Quantization as an Elementary Property of Macroscopic Vibrating Systems”, 21st Century Science & Technology, Vol. 18, No. 4, 50-63 (2006). [http://www.21stcenturysciencetech.com/2006_articles/Amplitude.W05.pdf] D.B.Doubochinski, J. Tennenbaum, On the Fundamental Properties of Coupled Oscillating Systems” (2007). arXiv:0712.2575v1 [physics.gen-ph] D.B. DoubochinskiI, J. Tennenbaum, “The Macroscopic Quantum Effect in Nonlinear Oscillating Systems: a Possible Bridge between Classical and Quantum Physics” (2007). arXiv:0711.4892v1 [physics.gen-ph] D.B. DoubochinskiI, J. Tennenbaum, “On the General Nature of Physical Objects and their Interactions as Suggested by the Properties of Argumentally-Coupled Oscillating Systems” (2008). arXiv:0808.1205v1 [physics.gen-ph]

26. Slide 26/25