1. Classical Mechanics
PHYS 2006
Tim Freegarde

2. Coupled oscillators

3. Coupled oscillators normal modes eigenmodes, eigenfrequencies
Ethan Neil, U Colorado
K Wayne Land
normal modes
eigenmodes, eigenfrequencies

4. Coupled oscillators beating when normal modes superposed
SYMMETRIC
ANTI-SYMMETRIC
beating when normal modes superposed
neutrino oscillations between flavours
T Kajita, A McDonald (Nobel 2015)
avoided crossings, ac Stark effect, …

5. Triatomic molecule
determine force on, hence acceleration of, each atom
seek harmonic solutions to
where
set , and
solutions are roots of
TRANSLATION
SYMMETRIC
ASYMMETRIC

6. Normal modes of a beaded string

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

8. 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

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

10. Avoided crossings NYQUIST SAMPLING THEOREM
BRILLOUIN ZONE
NYQUIST SAMPLING THEOREM
wavenumbers beyond Brillouin zone repeat motion of lower values
forward and backward travelling waves act as coupled oscillators
at , forward and backward travelling waves degenerate → avoided crossing

11. Coherent control: the pianoforte
soundboard b
keyboard
hammer
strings
bridge
escapement
action
strong coupling
 rapid decay
time
logP
weak coupling
 slow decay
MODES OF
COUPLED MOTION
logP
time
“PROMPT”
“AFTERSOUND”
strike strings together to produce almost pure s
strike one string - una corda - to produce equal superposition
strike strings together from opposite sides to produce almost pure a
reinforced “aftersound” is audible even when played softly
FORTE
PIANO
G Weinreich, Sci Am (Jan 1979) pp94-102

12. Weakly coupled oscillators

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

14. Classical Mechanics
PHYS 2006
Tim Freegarde