Class #23 of 30

1. Class #23 of 30 • Tennis racket demo • Euler’s equations • Motion of a free body • No external torques • Response to small disturbances • Tennis racket theorem • Nutation • Chandler wobble • Inertia tensor redux • Inertia tensor of a cube • Inertia tensor of a rectangle • Lamina theorem :02

2. Inertia Tensor • If choose principal axes, Inertia tensor is diagonal :02

3. Inertia Tensor – U-solve it Calculate elements of an inertia tensor for a rectangular lamina. B A

4. Vectors expressed in rotating frames For fixed e-sub-I For rotating e-sub-I Imagine same axes (x,y,z) expressed in two frames S0 stationary) and S (fixed to earth). :60

5. Euler’s Equation :60

6. Small perturbations :60

7. Small perturbations -II :60

8. Class #23 Windup - I • Euler • Without external torques • Rotations about the “middle-valued” principal axis are unstable :60

9. 23 11/12 Ch. 10 Euler’s equations – “tennis racket theorem” #12 – Supplement H, Taylor 9.11, 10.7, 10.18 11/14 Home stretch 24 11/14 Exam review + Ch. 11 Coupled Oscillators and Normal modes 11/21 25 11/19 Test #3 Central force and accelerated frames This exam was moved to 11/19 from 11/14 26 11/21 Coupled Oscillators / Nonlinear Mechanics and Chaos 11/26 27 11/26 Ch. 12 Nonlinear Mechanics and Chaos 12/5 11/28 THANKSGIVING 28 12/3 Ch. 13 Hamiltonian Mechanics - 29 12/5 Test #4 Inertia Tensor / Normal Coords / Chaos 30 12/10 Review problems for Final - :02

10. Class #23 Windup - II • Thursday bring problems you want to see worked • Homework is due day of exam (next Tuesday) – A few more problems will be added to assignment • Will go over old homeworks and your q’s in class Thursday :60

11. Class #24 of 30 • Exam -- Tuesday • Additional HW problems posted Friday (also due Tuesday). • Bring Index Card #3. • Office hours on Monday 3:30-6:00 • Topics • Central Force • Kepler’s Laws • Gravitational PE and Force • Small Oscillations about equilibrium • Reduced Mass • Momentum conservation • Pseudopotential :02

12. Class #24 of 30 • Central Force • Polar form of orbital equations • Elliptical orbits • Hyperbolic orbits • Parabolic / Circular Orbits • Scattering • Accelerated Reference Frames • Effective Gravity • Centrifugal force • Coriolis force :02

13. Problem Review • KKR9-4 Holy Earth • KKR9-6 Eccentric comet • KKR9-8 Escape to the moon • Taylor 8-9 Small oscillations • Taylor 9-1/9-2 Buoyant doughnuts • Taylor 9-9 Tilted Plum-line • Taylor 9-10 Spin the bucket :02

14. Gravity and Electrostatics Gravity Electrostatics Universal Constant Force Law Gauss’s Law Potential :08

15. Kepler’s 1st, 2nd and 3rd laws (1610) • 1st Law – Planets move in ellipses with sun at one focus • Third law demonstrated previously relates period to semi-minor radius • 2nd law is direct consequence of momentum conservation • “Equal areas are swept out in equal times” • True for ALL central forces :37

16. E, L and Eccentricity The physics is in E and L. Epsilon is purely a geometrical factor. Epsilon equation applies to ALL conic sections (hyperbolae, ellipses, parabolas). :30

17. Central Force <- Completely general for inverse square forces … All types of orbits. <- “Gamma” makes it specific for gravity. Key constants are, (E and L), OR (c and L) or (L and epsilon) or (c and epsilon) <- Specific for elliptical orbits. :30

18. Planetary Scattering Angle Sketch for epsilon=2 :37

19. Reduced two-body problem :15

20. Equivalent 1-D problem Relative Lagrangian Radial equation Total Radial Force :30

21. Class #24 Windup :60

22. :60

23. Class #26 of 30 • Nonlinear Systems and Chaos • Most important concepts • Sensitive Dependence on Initial conditions • Attractors • Other concepts • State-space orbits • Non-linear diff. eq. • Driven oscillations • Second Harmonic Generation • Subharmonics • Period-doubling cascade • Bifurcation plot • Poincare diagram • Feigenbaum number • Universality :02

24. Outline • Origins and Definitions of chaos • State Space • Behavior of a driven damped pendulum (DDP) • Non-linear behavior of a DDP • Attractor • Period doubling • Sensitive dependence • Bifurcation Plot :02

25. Definition of chaos • The dynamical evolution that is aperiodic and sensitively dependent on initial conditions. In dissipative dynamical systems this involves trajectories that move on a strange attractor, a fractal subspace of the phase space. This term takes advantage of the colloquial meaning of chaos as random, unpredictable, and disorderly behavior, but the phenomena given the technical name chaos have an intrinsic feature of determinism and some characteristics of order. • Colloquial meaning – disorder, randomness, unpredictability • Technical meaning – Fundamental unpredictability and apparent randomness from a system that is deterministic. Some real randomness may be included in real systems, but a model of the system without ANY added randomness should display the same behavior. :02

26. Weather and climate prediction • Is important • Can we go: • For a hike? • Get married outdoors? • Start a war? • Will we be hurt by a: • Tornado? • Hurricane? • Lightning bolt? • How much more fossil fuel can we burn before we: • Fry? • Drown? • Starve? :02

27. Early work by Edward N. Lorenz • 1960’s at MIT • Early computer models of the atmosphere • Were very simple • (Computers were stupid) • Were very helpful • Results were not reproducible!! • Lorenz ultimately noticed that • For 7-10 days of prediction, all of his models reproduced very well. • After 7-10 days, the same model could be run twice and give the same result – BUT!! • Changes that he thought were trivial • (e.g. changing the density of air by rounding it out at the 3th decimal place, or slightly modifying the initial conditions at beginning of model) • Produced COMPLETELY DIFFERENT results • This came to be called “The Butterfly effect” :02

28. State Space :02

29. Viscous Drag III – Stokes Law D Form-factor k becomes “D” is diameter of sphere Viscous drag on walls of sphere is responsible for retarding force. George Stokes [1819-1903]  (Navier-Stokes equations/ Stokes’ theorem) :45

30. Damped Driven Pendulum (DDP) • Damping • Pendulum immersed in fluid with Newtonian viscosity • Damping proportional to velocity (and angular velocity) • Driving • Constant amplitude drive bar • Connected to pendulum via torsion spring • Torque on Pendulum is :02

31. Damped Driven Pendulum (DDP) :02

32. Conditions for chaos • Dissipative Chaos • Requires a differential equation with 3 or more independent variables. • Requires a non-linear coupling between at least two of the variables. • Requires a dissipative term (that will use up energy). • Non-dissipative chaos • Not in this course :60

33. Sensitive dependence on initial conditions :60

34. Sensitive dependence on initial conditions :60

35. Worked problem • Sketch a state-space plot for the magnetic pendulum • Indicate the attractors and repellers • Show some representative trajectories • First explore the trajectories beginning with theta-dot=0 • Then explore trajectories that begin with theta in some state near an attractor but through proper choice of theta-dot move to the other attractor. • Sketch the basins of attraction :60

36. Class #26 Windup • Homework • Is on Chapters 9 and 10 – Read them! • Is partly posted now … more will be added tomorrow. • Is due BEFORE Thanksgiving (Wednesday … or under my door Thursday). :60

37. Class #27 Notes • Homework • Is due before you leave • Problem 10-25 has been upgraded to extra-credit. • Probs 10-21 and 10-24 are CORE PROBLEMS. Make sure you understand these. • When you return – • Will spend another lecture on tops, tensors and Euler’s theorem before the exam. :60

38. Class #27 of 30 • Nonlinear Systems and Chaos • Most important concepts • Sensitive Dependence on Initial conditions • Attractors • Other concepts • State-space orbits • Non-linear diff. eq. • Driven oscillations • Second Harmonic Generation • Subharmonics • Period-doubling cascade • Bifurcation plot • Poincare diagram • Mappings • Feigenbaum number • Universality :02

39. Chaos on the ski-slope 7 “Ideal skiers” follow the fall-line and end up very different places :60

40. Insensitive dependence on initial conditions :60

41. Sensitive dependence on initial conditions :60

42. Resampled pendulum data Gamma=1.077 Gamma=0.3 Gamma=1.0826 Gamma=1.105 :60

43. Bifurcation plot 0.3 +100 +50 0 -50 -100 :60

44. Bifurcation plot and universality • For ANY chaotic system, the period doubling route to chaos takes a similar form • The intervals of “critical parameter” required to create a new bifurcation get ever shorter by a ratio called the Feigenbaum #. :60

45. In and out of chaos :60

46. Poincare plot :60

47. Poincare plot • Poincare plot is set of allowed states at any time t. • States far from these points converge on these points after transients die out • Because it has fractal dimension, the Poincare plot is called a “strange attractor” :60

48. State-space of flows :60

49. Cooking with state-space • Dissipative system • The net volume of possible states in phase space ->0 • Bounded behavior • The range of possible states is bounded • The evolution of the dynamic system “stirs” phase space. • The set of possible states gets infinitely long and with zero area. • It becomes fractal • A cut through it is a “Cantor Set” :60

50. Mapping vs. Flow Gamma=1.0826 • A Flow is a continuous system • A flow moves from one state to another by a differential equation • Our DDP is a flow • A mapping is a discrete system. • State n-> State n+1 according to a difference equation • Evaluating a flow at discrete times turns it into a mapping • Mappings are much easier to analyze. Gamma=1.105