Classical Mechanics PHYS 2006 Tim Freegarde 2020 21

Classical Mechanics PHYS 2006 Tim Freegarde 2020 -21 27 Beaded string

Normal modes of a beaded string 2

Normal modes of a beaded string 3

Normal modes of a beaded string 4

Normal modes of a beaded string 5

Normal modes of a loaded spring 6

Normal modes of a beaded string AV E IT M I L W • band theory of semiconductors • photonic crystals • bonding in benzene 7

Normal modes of a beaded string BRILLOUIN ZONE AV E IT M I L FORWARD & BACKWARD WAVES DEGENERATE AVOIDED CROSSING W NYQUIST SAMPLING THEOREM: if ALIASING then AVOIDED CROSSING: • forward and backward waves act as coupled oscillators • at , forward and backward travelling waves degenerate → avoided crossing 8

Avoided crossings ω22 1 2 ANTI-SYMMETRIC • if • fix , oscillator frequencies and vary …somehow… 1 2 SYMMETRIC • when or , all mode energy all in one oscillator • frequency splitting depends upon coupling strength ω22 ω12 • when , antisymmetric or symmetric mode: energy divided equally 1 2 2 only ADIABATIC PASSAGE • sweep from to (or reverse) • complete energy transfer from one oscillator to the other 1 2 1 only ω12 ω22 9

Normal mode boundary conditions 10

Normal modes of a finite beaded string • determine force on, hence acceleration of, each bead • seek harmonic solutions to where • assume translational symmetry • dispersion relation 11

Classical Mechanics LINEAR MOTION OF SYSTEMS OF PARTICLES centre of mass Newton’s 2 nd law for bodies (internal forces cancel) rocket motion rotations and infinitessimal rotations ANGULAR MOTION angular velocity vector, angular momentum, torque parallel and perpendicular axis theorems rigid body rotation, moment of inertia, precession conservative forces, law of universal gravitation GRAVITATION & KEPLER’S LAWS 2 -body problem, reduced mass planetary orbits, Kepler’s laws energy, effective potential NON-INERTIAL REFERENCE FRAMES NORMAL MODES centrifugal and Coriolis terms Foucault’s pendulum, weather patterns coupled oscillators, normal modes boundary conditions, Eigenfrequencies 12

Classical Mechanics PHYS 2006 Tim Freegarde