Spring 2016
1. Useful Notions and Mathematics
2. Simple Harmonic Oscillators (SHOs)
3. Energy Conservation
4. SHOs and Near-Equilibrium Potential Energy
5. The superposition of SHOs
1/18/2016

0 th : Notions and Some Mathematics
Friction less
Friction
1/18/2016

Euler’s Complex Analysis
Complex number definition: x
2 i
2 i
 
1 And also satisfied by -i z
ˆ  x
 iy x & y : real numbers
Euler’s formula: e i
  cos
  i sin

1/18/2016
Leonhard Euler (1707-1783)

Let’s define
Then z
   i
 dz


Therefore
z
 i

"our jewel…one of the most remarkable, almost astounding, formulas in all of mathematics.“ Richard Feynman
1/18/2016

1/18/2016
Define z
ˆ  x
 iy z
ˆ  r (cos
  i

The conjugate z
ˆ
*
 x
 iy z
ˆ*  r (cos
  i
  e
 i


Rotating Arrow + Phase Angle
 t
1/18/2016
Representation of a complex number in terms of real and imaginary components
   t

Complex Representation
Phasor form:
1/18/2016
 t

use physics/mechanics to write equation of motion for system restoring force
F s
  s x ( t )
Inertial force F i d
2
 m dt
2 x ( t )
 m  x  ( t ) equation of motion m  x  ( t )
 sx ( t )

0 insert generic trial form of solution find parameter values for which trial form is a solution
 t
 x  ( t )
 
2 x ( t )

0
1/18/2016
  s m

Solution: Complex Representation
 x  ( t )
 
0
2 x ( t )

0 
0
 s m
Phasor form:
Real Part:
1/18/2016

Simple Harmonic Oscillators (SHOs)
X=0
X=A; v=0; a=-a max
X=0; v=-v max
; a=0
X=-A; v=0; a=a max
X=0; v=v max
; a=0
X=A; v=0; a=-a max
X=-A X=A
1/18/2016

Linearly Polarized Light as Phasors
E ( t )

E
0 e
 i
 t
 i

1/18/2016
Phase matters!!!

Linearly Polarized Light as Phasors
E ( t )

E
0 e
 i
 t
 i

1/18/2016
Phase matters!!!

Gravitational wave detected. Or not?
1/18/2016
The Laser Interferometer Gravitational-
Wave Observatory ( LIGO )

Other Model Systems Using SHOs
1/18/2016

Energy conservation of SHOs
Mechanical & Electrical Oscillator
1/18/2016

SHOs & Near-Equilibrium-Potential
Simple Pendulum
1/18/2016

SHOs & Near-Equilibrium-Potential
L-J potential for interaction between a pair of neutral atoms or molecules
1/18/2016

Arbitrary potentials V(x) can be well approximated by the power series near x
0 by Taylor expansion
Example: a Taylor series of a function f(x)
Near x
0
=0
1/18/2016

Adding Two Vibrations of Same Amplitudes
(I) Equal Frequencies
Two Parallel SHOs
1/18/2016

Additions of Two Parallel SHOs
(I) Equal Frequencies
1/18/2016

Equal Frequencies
1/18/2016
[ + ]
22

Adding Vibrations of Equal Frequency
Examples
1/18/2016

Superposition of Many SHOs
1/18/2016