1. 2. Nonlinear Systems. Part I 1.Nonlinear Mechanics 2.Competition Phenomena 3.Nonlinear Electrical Phenomena 4.Chemical and Other Oscillators

2. 2.1. Nonlinear Mechanics 1.The Simple Pendulum 2.The Eardrum 3.Nonlinear Damping 4.Nonlinear Lattice Dynamics

3. 2.1.1. The Simple Pendulum

4. Lagrangian approach : The file MF01.nb / MF01.mws attempts, but failed, to solve this analytically.MF01.nbMF01.mws Example 2-1: Parametric Excitation: Example 2-1.nb02-1.nb

5. 2.1.2. The Eardrum Harmonic force: After an initial transient period, respond only to  Forced simple harmonic oscillator

6. Eardrum is asymmetrically loaded Helmholtz’s eardrum equation, 1895

7. 2.1.3. Nonlinear Damping For an ellipsoidal object moving in a fluid without creating turbulence Equation of motion for an object moving near the earth’s surface Stoke’s Law of Resistence: n =1 Newton’s Law of Resistence : n = 2

8. Motion of a military shell in air 1.n ~ 1 for v  24 m/s or 86 km/h 2.n ~ 2 for 24 m/s < v < v s ( ~300 m/s ) 3.For v ≥ v s, there is a “bump” in the |F drag | vs |v| curve 4.n ~ 1 above ~600 m/s.

9. Nonlinear Air Drag on a Sphere: MF02.nbMF02.mws Drag and Lift on a Golf Ball: MF03.nbMF03.mws

10. 2.1.4. Nonlinear Lattice Dynamics E. Fermi, J.Pasta, S. Ulam: FPU problem N  64 (eardrum problem )

11. If springs are harmonic (   0 ), motion is linear combinations of normal modes. which conserve energies. A system of harmonic springs is NOT ergodic. ( can’t reach thermal equilibrium) FPU showed that their problem was also non-ergodic. This is known as the FPU anomaly. If  >  C, there is a transition into chaos.  equi-partition of energy, if not actual ergodicity.

12. Toda lattice

13. For small r, (FPU problem)