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Logistics

Logistics. Lecture notes allowed No numbers Same format as mid-terms. Four problems. I will try to mix several concepts in one problem! Formulas from front page of Taylor will be given If you feel you need some formula, don’t hesitate to ask!. How to prepare.

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Logistics

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  1. Logistics • Lecture notes allowed • No numbers • Same format as mid-terms. Four problems. I will try to mix several concepts in one problem! • Formulas from front page of Taylor will be given • If you feel you need some formula, don’t hesitate to ask!

  2. How to prepare • Review your lecture notes and make sure they are complete • Read the handouts that I posted • Solve all homework • Solve your mid-term tests • Solutions are posted, but don’t look at them before you solve the problem! • Work out examples in textbook and lecture notes, and look through end-of-chapter problems • Don’t hesitate to contact me if you have any difficulties

  3. Math • Vectors, dot and cross product; Levi-Civita symbol • Calculus • Integrate by substitution of variable • Differential in standard coordinate systems • Vector calculus (formulas will be given as needed) • Differential equations: • Solve by separation of variables • Solve linear equations • Solve harmonic oscillator equation • Apply initial conditions • Approximations, expansions, linearization

  4. Linear problems • Know the solution to the harmonic oscillator problem! No dissipation, with dissipation, driven, any initial conditions • Know how to solve a general system of linear ODE with constant coefficients!

  5. Bird’s view of the classical mechanics • Newton’s 2nd law: intuitive, but has different form in different coordinates • The only conservation laws that can be applied “automatically” are energy, momentum, or angular momentum Given a problem, now we know many ways to solve it

  6. Lagrangian approach • Based on variational principle; therefore invariant with respect to transformation of coordinates (q1, …qn)->(Q1, …,Qn)* • Equations have the same form in any coordinate system! • In practice, you still start from natural coordinates and L = K-U, and then make the transformation of your choice *Note: velocities are not transformed independently, they follow the transformation of q

  7. Lagrangian approach: what kind of problems can be solved? • All linear problems without explicit time dependence in coefficients. L contains only qiqj and terms (quadratic). Then E-L equations contain only linear terms. • Linear in term in the Lagrangian can be eliminated as a total derivative df/dt • T and U can be diagonalized simultaneously by transformation to normal frequencies and normal coordinates. Then the solution is straightforward. Expect such problem on the final exam.

  8. Coupled oscillations • Normal frequencies and normal modes • Normal coordinates: transformations in both directions • Inhomogeneous terms (e.g. constant force) • Typical problems: springs, pendulums

  9. Lagrangian approach: what kind of problems can be solved? • Nonlinear problem if • Only one degree of freedom and no time dependence, i.e. energy is conserved • Then equation E = T+U leads to the first-order equation for x(t) that can always be solved in quadratures • All other nonlinear problems can be solved only by reduction to a set of 1D problems

  10. Nonlinear problems • Need to have enough integrals of motion (equal to the number of degrees of freedom) • Within the Lagrangian approach, the only regular way of finding I.o.M. is through cyclic coordinates. Then the generalized momentum is conserved. • Choosing the right system of coordinates is critical! It should reflect the symmetry of the potential U(qi) • Examples: central force, EM field

  11. Nonlinear problems • Even when the problem is not solvable, there is a powerful qualitative method: phase space • Motion is completely described by finding critical (stationary) points and trajectories in their vicinity • Motion in the vicinity of stationary points is described by linearized equations! Know how to linearize!! • Only four types of points on a 2D phase plane • Know how to find them and the structure of phase trajectories around them!

  12. Hamiltonian dynamics • Also based on the variational principle, same invariance with respect to coordinate transformations • Know how to go from L to H! (next slide) • Motion as a flow of “phase fluid” in phase space • Powerful theorems and methods: • Liouville’s theorem (conservation of phase volume) • Poisson’s brackets: generate evolution of any function f(p,q), may give new integrals of motion, allow you to check if the transformation is canonical

  13. Notable exceptions from H = K+U rule • Non-inertial reference frame, for example polar coordinates for a bead on a rotating wire • Charged particle in an electromagnetic field

  14. Canonical transformations • The greatest advantage! Transform both q and p. Much broader class than purely coordinate transformations • The main approach: transform to new canonical coordinates in which the solution to Hamilton’s equations is trivial • Know how to find the generating function and write the solution in old coordinates

  15. Hamilton-Jacobi theory • Most general way of finding integrals of motion: by separation of variables in HJ equation • Again, identifying the right system of coordinates is critical • Method in search of problems: first identify potentials and coordinates for which variables can be separated, then see what problems it can solve

  16. Know the procedure! • Choose the coordinates • Know the Hamiltonian in standard coordinate systems! • Separate variables • Write the solution • Apply initial conditions

  17. Rigid body rotation • Don’t forget PHYS218 stuff! • Center of mass • Angular momentum and it’s split into COM and “about COM” motion • Kinetic energy and it’s separation into KE of COM + KE of the motion relative to COM • Rotation about fixed axis z. Moment of inertia. Lz = IzΩ. Polar coordinates.

  18. Rigid body rotation • L is not parallel to Ω. Tensor of inertia. • Diagonalization of TI. Principal moments and axes. Know how to do this! • Euler’s equations • Symmetric top • Lagrangian and Hamiltonian in terms of Euler’s angles • Symmetric top

  19. Non-separable (non-integrable) systems • Only approximate methods • Adiabatic invariants • Know how to find the actions and convert to action-angle variables • Know how to use adiabatic invariants to find the time dependence of energy, amplitude, and other physical quantities • Method of averaging (not needed for 2014)

  20. Road to chaos • Nonlinearity • Non-integrability • Resonances • Destruction of separatrices

  21. Resonances • Nonlinear resonance and nonlinear pendulum • Back to phase space • Instability and destruction of a separatrix • Bifurcations • Period-doubling bifurcation • Properties of chaos • From laminar to turbulent phase flow

  22. Bifurcation diagram of a driven pendulum θ(tn) γ

  23. Poincare section

  24. Poincare section of the phase plane of a driven pendulum http://www.physik3.gwdg.de http://www.elmer.unibas.ch/pendulum/chaos.htm

  25. wikipedia

  26. Poincare maps on y,dy/dt plane for Henon-Heiles problem C. Emanuelsson

  27. Self-similarity C. Emanuelsson

  28. KAM tori for Henon-Heiles problem C. Emanuelsson

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