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A Classical Mechanical Road to Relativity and Quantum Theory
Honors Physics Colloquium
2016 Spring PHYS 3922H

2016 Detailed List of Lecture Topics    

Lectures 1 & 2.  1st axioms and theorems of classical mechanics I. & II.    (1/19/2016 View on YouTube & 1/21/2016 View on YouTube)

(Ch. 1 and Ch. 2 of Unit 1)


Geometry of momentum conservation axiom (ala Occam’s Razor)

Totally Inelastic “ka-runch”collisions* (begin 4:1 graph project)

Perfectly Elastic “ka-bong” and Center Of Momentum (COM) symmetry*

+Intro to weighty-averages and vector notation

Comments on idealization in classical models


Geometry of Galilean translation symmetry

45° shift in (V1,V2)-space

Time reversal symmetry

... of COM collisions


Algebra, Geometry, and Physics of momentum conservation axiom

Vector algebra of collisions

Matrix or tensor algebra of collisions

Deriving Energy Conservation Theorem


Numerical details of collision tensor algebra


2016 Honors Physics Colloquium web site       Link → http://www.uark.edu/ua/modphys/markup/HPColloqWeb.html
* Launch Vehicle Collision Simulator       Link → http://www.uark.edu/ua/modphys/markup/CMMotionWeb.html
* Launch Superball Collision Simulator       Link → http://www.uark.edu/ua/modphys/markup/BounceItWeb.html
Graph Paper: R for display on right side (i.e. Even numbered pages)
      SUVVW To Fit 8x10   SUVVW To Fit 8x10R   SUV VW 240x80   SUVVW 240x80R


Lectures 3 & 4.  Analysis of 1D 2-Body Collisions I. & II.    (1/26/2016 View on YouTube & 1/28/2016 View on YouTube)

(Ch. 2 to Ch. 4 of Unit 1)


Review of elastic Kinetic Energy ellipse geometry


The X2 Superball pen launcher*

Perfectly elastic “ka-bong” velocity amplification effects (Faux-Flubber)


Geometry of X2 launcher bouncing in box

Independent Bounce Model (IBM)

Geometric optimization and range-of-motion calculation(s)

Integration of (V1,V2) data to space-time plots (y1(t),t) and (y2(t),t) plots

Integration of (V1,V2) data to space-space plots (y1, y2)

Examples: (M1=7, M2=1) and (M1=49, M2=1)


Multiple collisions calculated by matrix operator products

Matrix or tensor algebra of 1-D 2-body collisions


Ellipse rescaling-geometry and reflection-symmetry analysis

Rescaling KE ellipse to circle

How this relates to Lagrangian, l’Etrangian, and Hamiltonian mechanics in Ch. 12


Graph Paper: Fine Protractor


Lecture 3-4 (Extended).  Analysis of 1D 2-Body Collisions III.    (2/2/2016 View on YouTube)

(Ch. 2 to Ch. 4 of Unit 1)


Review of elastic Kinetic Energy ellipse geometry


The X2 Superball pen launcher*

Perfectly elastic “ka-bong” velocity amplification effects (Faux-Flubber)


Geometry of X2 launcher bouncing in box

Independent Bounce Model (IBM)

Geometric optimization and range-of-motion calculation(s)

Integration of (V1,V2) data to space-time plots (y1(t),t) and (y2(t),t) plots

Integration of (V1,V2) data to space-space plots (y1, y2)

Examples: (M1=7, M2=1) and (M1=49, M2=1)


Multiple collisions calculated by matrix operator products

Matrix or tensor algebra of 1-D 2-body collisions


Ellipse rescaling-geometry and reflection-symmetry analysis

Rescaling KE ellipse to circle

How this relates to Lagrangian, l’Etrangian, and Hamiltonian mechanics in Ch. 12

Reflections in the clothing store: “It’s all done with mirrors!”

Introducing hexagonal symmetry D6~C6v (Resulting for m1/m2=3)

Group multiplication and product table

Classical collision paths with D6~C6v (Resulting from m1/m2=3)

     Other not-so-symmetric examples: m1/m2=4 and m1/m2=7 and (M1=100, M2=1)


Lecture 5 & 6.  Kinetic Derivation of 1D Potentials and Force Fields I & II.    (2/4/2016 View on YouTube) & (2/9/2016 View on YouTube)

(Ch. 5 of Unit 1)


Review of (V1,V2)→(y1,y2) relations       High mass ratio M1/m2 = 49


Force “field” or “pressure” due to many small bounces

Force defined as momentum transfer rate

The 1D-Isothermal force field F(y)=const./y and the 1D-Adiabatic force field F(y)=const./y3


Potential field due to many small bounces

Example of 1D-Adiabatic potential U(y)=const./y2

Physicist’s Definition F=-ΔU/Δy     vs.     Mathematician’s Definition F=+ΔU/Δy

Example of 1D-Isothermal potential U(y)=const. ln(y)


“Monster Mash”classical segue to Heisenberg action relations

Example of very very large M1 ball-wall(s) crushing a poor little m2

How does m2 conserve action (ΔxΔp or ∫p⋅dx) as its KE changes?

Lecture 7 & 8  Introducing Lightwave Fourier Analysis I & II    (2/9/2016 View on YouTube  &  (2/10/2016 View on YouTube)

(Comparing wave dynamics to classical behavior in Ch. 3 thru Ch. 5 of Unit 1)


Introducing lightwave Fourier analysis - Pulse Waves (PW) versus Continuous Waves (CW)

Simplest is CW (Continuous Wave, Cosine Wave, Colored Wave, Complex Wave,...)

CW parameters: Wavelength λ and Wave period τ

CW inverse parameters: Wavelnumber κ=1/λ and Wave frequency υ =1/τ

CW angular parameters: Wavevector k =2πκ=2π/λ and angular frequency ω =2πυ =2π/τ

CW wavefunction : ψ=A exp[i(kx-ωt)]= A cos(kx-ωt)+iA sin(kx-ωt)


Wave phasors, phasor chain plots, dispersion functions ω(k), and phase velocity Vphase=ω(k)/k

Special case: Lightwave linear dispersion:Vphase=c or: ω(k)=ck


Introducing PW (Pulse Wave, Particle-like Wave, Packet Wave,...) archetypes compared to CW

Building PW from CW components using “Fourier Control” app-panel

Fourier PW “box-car” geometric series summed

Animation of PW obeying lightwave linear dispersion ω(k)=ck

     Important Evenson axiom for relativity: “All go c”

Visualizing PW wave uncertainty relations for space: Δx⋅Δκ=1 and time: Δt⋅Δυ=1

PW “wrinkles” go away if Fourier “boxcar” is tapered to a softer “Gaussian”


Opposite-pair CW (colliding ±m=±2) Fourier components trace a Cartesian space-time grid


Lecture 9.  Fractal behavior in matter-wave “Tiny-Big-Bang” (2/16/2016)   View on YouTube

(Quantum wave analogies to classical “Monster-Mash” model in Ch. 5 of Unit 1)


Reviewing lightwave Fourier analysis - Pulse Waves (PW) versus Continuous Waves (CW)

Opposite-pair CW (colliding ±m=±2) Fourier components trace a Cartesian space-time grid

Colliding PW lightwaves trace space-time “baseball diamonds”

Introducing CW (colliding ±m=±2) Doppler shifted to (m=-1 and m=+4)


NON-Lightwaves whose ω(k) dispersion functions are NOT straight lines

Animating PW made of CW that have quadratic (Bohr-Schrodinger) dispersion

Visualizing PW wave uncertainty relations for space: Δx⋅Δκ=1 and time: Δt⋅Δυ=1

Matter-wave fractal behavior in a “Tiny-Big-Bang”

[Harter, J. Mol. Spec. 210, 166-182 (2001)];
[Harter, Li IMSS (2012)]

A lesson in geometry of fractions and fractals: Ford Circles and Farey Sums

[Lester. R. Ford, Am. Math. Monthly 45,586(1938)];
[John Farey, Phil. Mag.(1816) Wolfram];
[Li, Harter, Chem.Phys.Letters (2015]


Lecture 10.  Dynamics of Potentials and Force Fields (2/18/2016)   View on YouTube

(Ch. 6 and part of Ch. 7 of Unit 1)



Potential energy geometry of Superballs and related things

Thales geometry and “Sagittal approximation”

Geometry and dynamics of single ball bounce

(a) Constant force F=-k (linear potential V=kx )

     Some physics of dare-devil diving 80 ft. into kidee pool

(b) Linear force F=-kx (quadratic potential V=½kx2 (like balloon))

(c) Non-linear force (like superball-floor or ball-bearing-anvil)


Geometry and potential dynamics of 2-ball bounce

A parable of RumpCo. vs CrapCorp. (introducing 3-mass potential-driven dynamics)

A story of USC pre-meds visiting Whammo Manufacturing Co.


Geometry and dynamics of n-ball bounces

Analogy with shockwave and acoustical horn amplifier

Advantages of a geometric m1, m2, m3,... series

A story of Stirling Colgate (Palmolive) and core-collapse supernovae


Many-body 1D collisions

Elastic examples: Western buckboard

Bouncing columns and Newton’s cradle

Inelastic examples: “Zig-zag geometry” of freeway crashes

Super-elastic examples: This really is “Rocket-Science”

Lecture 11.   Geometry of common power-law potentials I. (2/23/2016)   View on YouTube

(Ch. 8 of Unit 1)


Geometry of common power-law potentials

Geometric (Power) series

“Zig-Zag” exponential geometry

Projective or perspective geometry

Parabolic geometry of harmonic oscillator kr2/2 potential and -kr1 force fields

Coulomb geometry of -1/r-potential and -1/r2-force fields

Compare mks units of Coulomb Electrostatic vs. Gravity


Geometry of idealized “Sophomore-physics Earth”

Coulomb field outside Isotropic & Harmonic Oscillator (IHO) field inside

Contact-geometry of potential curve(s)

“Crushed-Earth” models: 3 key energy “steps” and 4 key energy “levels”

Earth matter vs nuclear matter:

      Introducing the “neutron starlet” and  “Black-Hole-Earth”


Introducing 2D IHO orbits and phasor geometry

Phasor “clock” geometry

Graph Paper:
      Cartesian

Lecture 12.  Kepler Geometry of IHO (Isotropic Harmonic Oscillator) Elliptical Orbits (2/25/2016)   View on YouTube

(Ch. 8 and Ch. 9 of Unit 1)


“Sophomore-Physics-Earth” models: 3 key energy “steps” and 4 key energy “levels”

“Crushed-Earth” models: 3 key energy “steps” and 4 key energy “levels”

Earth matter vs nuclear matter:

     Introducing the “neutron starlet”

     Fantasizing a “Black-Hole-Earth


Introducing Isotropic Harmonic Oscillator (IHO) energy and frequency relations

Constructing 2D-IHO orbits using Kepler anomaly plots

Mean-anomaly and eccentric-anomaly geometry with web-app animation

Calculus and vector geometry of IHO orbits

Constructing 2D-IHO orbits using orbital phasor-clock plots

Phasor geometry of coordinate (x,y) and velocity (Vx,Vy) space with web-app animation

Kepler“laws” (Some that apply to all central (isotropic) F(r) force fields)

Angular momentum invariance of IHO: F(r)=-k·r with U(r)=k·r2/2

Angular momentum invariance of Coulomb: F(r)=-GMm/r2 with U(r)=-GMm·/r

Total energy E=KE+PE invariance of IHO: F(r)=-k·r

Total energy E=KE+PE invariance of Coulomb: F(r)=-GMm/r2

Link → BoxIt - Simulation of IHO Orbits w/ time rates of change
Link → RelaWavity - Geometry of IHO orbits w/ time rates of change
Link → RelaWavity - Geometry of IHO Ellipse Exegesis

Lecture 12. (Supplemental) Kepler Geometry of IHO (Isotropic Harmonic Oscillator) Elliptical Orbits (Advanced)  


Lecture 13 & 14  Complex Variables, Series, and Field Coordinates I & II    (3/1/2016 View on YouTube  &  (3/3/2016 View on YouTube)

(Ch. 10 of Unit 1)


1. The Story of e (A Tale of Great $Interest$)

How good are those power series?

Taylor-Maclaurin series, imaginary interest, and complex exponentials


2. What good are complex exponentials?

Easy trig

   Easy 2D vector analysis

      Easy oscillator phase analysis

         Easy rotation and “dot” or “cross” products


3. Easy 2D vector calculus

   Easy 2D vector derivatives

   Easy 2D source-free field theory

      Easy 2D vector field-potential theory


4. Riemann-Cauchy relations (What’s analytic? What’s not?)

Easy 2D curvilinear coordinate discovery

Easy 2D circulation and flux integrals

   Easy 2D monopole, dipole, and 2n-pole analysis

      Easy 2n-multipole field and potential expansion

         Easy stereo-projection visualization

            Cauchy integrals, Laurent-Maclaurin series


5. Mapping and Non-analytic 2D source field analysis

1. Complex numbers provide "automatic trigonometry"

2. Complex numbers add like vectors.

3. Complex exponentials Ae-iωt track position and velocity using Phasor Clock.

4. Complex products provide 2D rotation operations.

5. Complex products provide 2D “dot”(•) and “cross”(x) products.

6. Complex derivative contains “divergence”(∇•F) and “curl”(∇xF) of 2D vector field

7. Invent source-free 2D vector fields [∇•F=0 and ∇xF=0]

8. Complex potential φ contains “scalar”( F= ∇Φ) and “vector”( F=∇xA) potentials. The half-nʼ-half results: (Riemann-Cauchy Derivative Relations)

9. Complex potentials define 2D Orthogonal Curvilinear Coordinates (OCC) of field

10. Complex integrals ∫ f(z)dz count 2D “circulation”( ∫F•dr) and “flux”( ∫Fxdr)

11. Complex integrals define 2D monopole fields and potentials

12. Complex derivatives give 2D dipole fields

13. More derivatives give 2D 2N-pole fields…

14. ...and 2N-pole multipole expansions of fields and potentials...

15. ...and Laurent Series...

16. ...and non-analytic source analysis.


Lecture 15.  Introduction to classical oscillation and resonance (3/8/2016)   View on YouTube

(Ch. 2 of Unit 2)


1D forced-damped-harmonic oscillator equations and Green’s function solutions

Linear harmonic oscillator equation of motion.

Linear damped-harmonic oscillator equation of motion.

Frequency retardation and amplitude damping.

Figure of oscillator merit (the 5% solution 3/Γ and other numbers)

Linear forced-damped-harmonic oscillator equation of motion.

Phase lag and amplitude resonance amplification

Figure of resonance merit: Quality factor q=ω0/2Γ


Properties of Green’s function solutions and their mathematical/physical behavior

Transient solutions vs. Steady State solutions


Complete Green’s Solution for the FDHO (Forced-Damped-Harmonic Oscillator)

Quality factors: Beat, lifetimes, and uncertainty


Approximate Lorentz-Green’s Function for high quality FDHO (Quantum propagator)

Common Lorentzian (a.k.a. Witch of Agnesi)

Smith Chart

Lecture 16.  Introduction to coupled oscillation and eigenmodes I. (3/10/2016)   View on YouTube

(Ch. 3-4 of Unit 2)


Review of 1D FDHO (Forced-Damped-Harmonic Oscillator) response


2D harmonic oscillator (2D-HO) equations of motion

Lagrangian and matrix forms


2D harmonic oscillator equation eigensolutions (normal modes)

Eigensolutions by geometry for 2D-HO with bilateral (B-Type) symmetry

Symmetric (low frequency) mode versus antisymmetric (high frequency) mode

Mixed mode beat dynamics (with constant π/2 phase-lag)


Eigensolutions by matrix-algebra with example M = Matrix

Secular equation

Hamilton-Cayley equation and projectors

Idempotent projectors (how eigenvalues ⇒ eigenvectors)

Operator orthonormality and Completeness (Idempotent means: P·P=P)


Spectral Decompositions

Functional spectral decomposition

Orthonormality vs. Completeness vis-a`-vis Operator vs. State

Lagrange functional interpolation formula

Diagonalizing Transformations (D-Ttran) from projectors


Lecture 17.  Introduction to coupled oscillation and eigenmodes II. (3/15/2016)   View on YouTube

(Ch. 3-4 of Unit 2)


Review of 1D FDHO (Forced-Damped-Harmonic Oscillator) response


2D harmonic oscillator (2D-HO) equations of motion

Lagrangian and matrix forms


2D harmonic oscillator equation eigensolutions (normal modes)

2D classical HO compared to U(2) quantum 2-state system

Introducing ABCD Hamilton Pauli spinor symmetry expansion

Eigensolutions by geometry for 2D-HO with bilateral (B-Type) symmetry

Symmetric (low frequency) mode versus antisymmetric (high frequency) mode

Mixed mode beat dynamics (with constant π/2 phase-lag)

Geometry of phase and polarization


Eigensolutions by matrix-algebra with example M = Matrix

Secular equation

Hamilton-Cayley equation and projectors

Idempotent projectors (how eigenvalues ⇒ eigenvectors)

Operator orthonormality and Completeness (Idempotent means: P·P=P)


Spectral Decompositions

Functional spectral decomposition

Orthonormality vs. Completeness vis-a`-vis Operator vs. State

Lagrange functional interpolation formula

Diagonalizing Transformations (D-Ttran) from projectors


Video of Coupled Pendula

Lecture 17 (Extra)  Matrix algebra of quantum 2-state eigenmodes and dynamics (3/xx/2016)  

(Ch. 3-4 of Unit 2)


Lecture 18.  Mechanical analogs of quantum 2-state eigenmodes and dynamics (3/17/2016)   View on YouTube

(Ch. 3-4 of Unit 2)


Review of 2D classical HO compared to U(2) quantum 2-state system

Introducing ABCD Hamilton Pauli spinor symmetry expansion

Algebra of Hamilton/Pauli hypercomplex operators {σA,σB,σC}={σZ,σX,σY}

σA-products 3D vector analysis and “Crazy-Thing-Theorem”


Eigensolutions by matrix-algebra with example M = Matrix

Secular equation

Hamilton-Cayley equation and projectors

Idempotent (P·P=P) projectors (how eigenvalues ⇒ eigenvectors)

Eigenvector orthonormality and completeness

Spectral Decompositions

unctional spectral decomposition


U(2)⊃C2 ABCD group theory method to find 2D-HO eigenmodes and eigenvalues

Asymmetric-diagonal (AD-Type) symmetry

Bilateral-balanced (B-Type) symmetry

Circular-chiral-cycloton (C-Type) symmetry

Mixed ABCD symmetry examples


More theory of matrix diagonalization

Lagrange functional interpolation formula

Diagonalizing Transformations (D-Ttran) from projectors



Lecture 19 (Extra Topics)  Spinor-Vector relations and 2D-HO polarization dynamics (3/22/2016 Spring Break)  

(Ch. 4 of Unit 2)


Review: 2D harmonic oscillator equations with Lagrangian and matrix forms

ANALOGY: 2-State Schrodinger: iħ∂t|Ψ(t)〉=H|Ψ(t)〉 versus Classical 2D-HO: ∂2tx=-K•x

Hamilton-Pauli spinor symmetry ( σ-expansion in ABCD-Types) H=ωμσμ

Derive σ-exponential time evolution (or revolution) operator U = e-iHt = e-iσµωµt

Spinor arithmetic like complex arithmetic

Spinor vector algebra like complex vector algebra

Spinor exponentials like complex exponentials ("Crazy-Thing"-Theorem)

Geometry of evolution (or revolution) operator U = e-iHt = e-iσµωµt

The "mysterious" factors of 2 (or 1/2): 2D Spinor vs 3D Spin Vector space

     2D Spinor vs 3D vector rotation

     NMR Hamiltonian: 3D Spin Moment m in B field

Euler's state definition using rotations R(α,0,0), R(0,β,0),and R(0,0,γ)

Spin-1 (3D-real vector) case

Spin-1/2 (2D-complex spinor) case

3D-real Stokes Vector defines 2D-HO polarization ellipses and spinor states

Asymmetry SA=SZ, Balance SB =SX, and Chirality SC =SY

Polarization ellipse and spinor state dynamics

The "Great Spectral Avoided-Crossing" and A-to-B-to-A symmetry breaking



















See also: QTCA Lect. 9(2.12) p.61-103 for polarization ellipsometry


Lecture 20 (Extra)  Parametric Resonance and Multi-particle Wave Modes (3/xx/2016)  

(Ch. 7-8 of Unit 4 11.24.15)


Lecture 21.  CN Wave Modes (3/29/2016)   View on YouTube

(Ch. 5 of Unit 4 3.29.15)


Wave resonance in cyclic CN symmetry

Harmonic oscillator with cyclic C2 symmetry

C2 symmetric (B-type) modes

Projector analysis of 2D-HO modes and mixed mode dynamics

½-Sum-½-Diff-Identity for resonant beat analysis

½-Sum-½-Diff-Identity for resonant beat analysis

     Mode frequency ratios and continued fractions

     Geometry of that 90°-phase lag (again)

Harmonic oscillator with cyclic C3 symmetry

C3 symmetric spectral decomposition by 3rd roots of unity

Deriving C3 projectors

     Deriving and labeling moving wave modes

     Deriving dispersion functions and degenerate standing waves

          Examples by WaveIt animation

C6 symmetric mode model: Distant neighbor coupling

C6 moving waves and degenerate standing waves

C6 dispersion functions for 1st, 2nd, and 3rd-neighbor coupling

C6 dispersion functions split by C-type symmetry(complex, chiral, ...)


C12 and higher symmetry mode models: Archetypes of dispersion functions and 1-CW phase velocity

½-Sum-½-Diff-theory of 2-CW group and phase velocity


Lecture 21 (Extra)  CN-Symmetric Wave Modes (3/xx/2016)  

(Ch. 5 of Unit 4 3.29.15)


Lecture 22.  CN-Symmetric Wave Modes and 2-CW Algebra and Geometry (3/31/2016)   View on YouTube

(Ch. 5 of Unit 4)


Wave resonance in cyclic CN symmetry

C6 symmetric mode model: Distant neighbor coupling

C6 moving waves and degenerate standing waves

C6 dispersion functions for 1st, 2nd, and 3rd-neighbor coupling

C6 dispersion functions split by C-type symmetry(complex, chiral, ...)


C12 and higher symmetry mode models: Archetypes of dispersion functions and 1-CW phase velocity

½-Sum-½-Diff-theory of 2-CW group and phase velocity


Given two 1-CW phases find 2-CW phase velocity Vphase(2-CW) and group velocity Vgroup (2-CW)

Example: Bohr Dispersion 2-CW made of 1-CW(m=-1) and 1-CW(m=2)

2-CW space-time (x,t) lattice from per-space-time (κ,υ) by algebra

Same Example

Lecture 23.  : Relativistic wave mechanics I. - 1st-order Doppler shifts (4/5/2016)   View on YouTube

(Unit 3 4.05.16)


Special Relativity and Quantum Mechanics regarded as mysterious and lacking clarity

Bob & Alice regard for clarity of SR: foggy or QM: opaque

Can this situation be improved at fundamental axiomatic level?


Evidence and concepts needing critical review:

QM (Planck, 1900) and SR (Einstein, 1905) are both about light (em waves)

Galilean relativity, how it fails for light and how it doesn’t

     The great light-wave speed-limit (c=2.99792458m/s. by Evenson,...,Hall 1972)


Need better axioms (Occam’s Razors & Evenson’s Lasers): CW axioms outwit old PW axioms

Introduce “Keyboard of the gods” CW per-space-time (к,υ) that rules (λ,τ) space-time

Introduce idea of quantized wavenumber-кn and amplitude An (1st and 2nd quantization)

     Introduce infrared (IR) 300 THz, green 600THz, and ultra-violet (UV) 1200THz CW laser beams


Optical Doppler CW frequency shift υAB: A hidden key to understanding modern physics

Bob and Alice deduce Evenson’s CW Axiom: All march together at c = υλ = υ/к

Bob, Alice, and Carla discover rapidity (ρAB=ln υAB), a longitudinal measure of speed

     Bob, Alice, and Carla get Galileo’s Revenge Part I.: ρCBCAAB , a simple speed sum


Bob, Alice, and Carla get Galileo’s Revenge Part II.:and map space-time by phase-group 2-CW

½-sum-½-difference of phasor angular velocity determines space-time geometry

Relating rapidity ρAB and relativity velocity parameter βAB=uAB/c

Lecture 24.  : Relativistic wave mechanics II. 2nd-order effects (4/7/2016)   View on YouTube

(Unit 3 4.05.16)


Review of Doppler-shift and Rapidity ρAB calculation: Galileo’s Revenge Part I Lect. 23 p.64-75

Relating rapidity ρAB and relativity velocity parameter βAB=uAB/c


Review of ½-sum-½-difference Phase and Group factors giving relativistic space-axes and time-axes

Colliding-CW space-time (x,ct)-graph vs Colliding PW space-time (R,L)-baseball diamond


Review of ½-sum-½-difference of phasor angular velocity:Galileo’s Revenge Part II (Pirelli site)

Elementary models: 2-comb Moire′ patterns and cosine-law constructions


Bob, Alice, and Carla combine Doppler shifted ½-sum-½-difference Phase and Group factors

Doppler shifted Phase vector P′ and Group vector G′ in per-space-time

Minkowski coordinate grid in space-time

Animations that compare Doppler shifted colliding CW with colliding PW


The 16 parameters of Doppler-shifted 2-CW Minkowski geometry

Doppler shifted Phase parameters

Doppler shifted Group parameters

Lorentz transformation matrix and Two Famous-Name Coefficients


Thales Mean Geometry (Thales of Miletus 624-543 BCE) and its role in Relawavity

Detailed geometric construction of relawavity plot for 1-octave Doppler (βAB=uAB/c=3/5)


Stellar aberration and the Epstein approach to SR


Lecture 25.  : Relativistic wave mechanics III. 2nd-order effects (4/12/2016)   View on YouTube

(Unit 3 4.12.16)


Review: Rapidity ρ=ρAB, Doppler shifts e±ρ, and SR velocity parameter Vgroup/c=βAB=uAB/c=tanhρAB

Geometric construction steps 1-4 : 1-octave Doppler (e=2, e=1/2), (βAB=uAB/c=3/5)

Reviewing wave coefficients we’ll need to know (backwards and forwards)


Comparison of group and phase dynamics: FAST(er) (β=u/c=3/5) vs SLOW(er) (β=u/c=1/5)


Thales Mean Geometry (Thales of Miletus 624-543 BCE) and its role in Relawavity

Geometric construction steps 5,6,...: Per-space-time (ω,ck) dispersion hyperbola ω = Bcoshρ...

A quick flip to space-time (ct,x) construction: Minkowski coordinate grid


Lorentz transformations of Phase vector P′ and Group vector G′ in per-space-time

Lorentz matrix transformation of (x,ct) space-time coordinates

Two Famous-Name Coefficients: Lorentz space contraction and Einsein time dilation

Heighway Paradoxes: A relativistic “He said-She-said” argument


Phase invariance...derives Lorentz transformations

Another view: phasor-invariance and proper time


Yet another view: The Epstein space-proper-time approach to SR


Lecture 26.  : Relativistic wave mechanics IV. Coordinate geometry (4/14/2016)   View on YouTube

(Unit 3 4.14.16)


Review of geometric construction , per-space-time (ω,ck) dispersion hyperbola ω = Bcoshρ...

A quick flip to space-time (ct,x) construction: Minkowski coordinate grid


Lorentz transformations of Phase vector P′ and Group vector G′ in per-space-time

Lorentz matrix transformation of (x,ct) space-time coordinates

Two Famous-Name Coefficients: Lorentz space contraction and Einsein time dilation

Heighway Paradoxes: A relativistic “He said-She-said” argument


Phase invariance...derives Lorentz transformations...and vice-versa

Another view of phasor-invariance

Geometry of invariant hyperbolas

     Algebra of invariant hyperbolas

          Proper time τ0 and proper frequency ω0

               A politically incorrect analogy of rotation to Lorentz transformation


Yet another view: The Epstein space-proper-time approach to SR uses stellar aberration angle σ

Relating rapidity ρ to stellar aberration angle σ and circular or hyperbolic arc-area

Each circular trig function has a hyperbolic “country-cousin” function


Ship vs Lighthouse sagas and the Bureau of Inter-Galactic Aids to Navigation at Night (Our 1st RelativIt animations).

Lecture 27.  : Relativistic wave mechanics V. Coordinate geometry and Applications I (4/19/2016)   View on YouTube

(Unit 3 p.19-32 - 4.19.16)


Ship vs Lighthouse sagas and the Bureau of Inter-Galactic Aids to Navigation at Night (Our 1st RelativIt animations).

2005 and 2016 animations of lighthouses and ships in (x,y) scenarios and Minkowski (x,ct) plots

Lighthouse (x,y) frame: Dual concentric circular wavefronts serve as timing device

     Ship frame: time dilation Δ=coshρ=1.15 of Lighthouse blinks

          Simultaneous events in Lighthouse (x,y) frame: Not so in Ship (x′,y′) frame

               Simultaneous events in Lighthouse (x,y) frame: Not so in Ship (x′,y′) frame


Overlapped Lighthouse (x,ct) and Ship (x′,ct′) frame Minkowski plots correlate inconsistencies

Ship (x′,y′) frame: Dual un-concentric circular wavefronts map space-time

Pythagorean derivation of time-dilation factor Δ=coshρ

     Un-concentric derivation of stellar aberration k-angle σ


Per-spacetime 4-vector (ω0xyz) =(ω,ckx,cky,ckz) transformation

“Occam-sword” geometry: A pattern recognition aid

Relating velocity parameter β=u/c to rapidity ρ to k-angle σ to u/c-angle ν

     Circular arc-area σ vs. hyperbolic arc-area ρ

          Each circular trig function has a hyperbolic “country-cousin” function


Yet another view: The Epstein space-proper-time approach to SR uses stellar aberration k-angle σ

Lecture 28.  : Relativistic wave mechanics VI. Velocity geometry and Applications II (4/21/2016)   View on YouTube

(Unit 3 p.28-42 - 4.21.16)


A neo-liberal trigonometry lesson (sine, tangent, and secant) functions of angular sector area σ

Complimentary functions (... cosine, and cotangent, cosecant)

Hyper-trigonometry of ( tanhρ, sinhρ, and coshρ, sechρ, and cschρ, cothρ )

     Functions of hyper-angular sector area ρ related to functions of σ

          Each circular trig function has a hyperbolic “country-cousin” function

               ...and big-party fun was had by all!


Pattern recognition aids and “Occam-sword” geometry

Relating velocity parameters β=u/c to rapidity ρ to k-angle σ to u/c-angle ν

Relating wave dimensional parameters of phase wave and group wave

     Parameter-space symmetry points


Yet another view: The Epstein space-proper-time approach to SR uses stellar aberration k-angle σ

Review of proper time relations and basis of Epstein’s cosmic speedometer

Epstein geometry for relativistic parameters


Spectral details of per-spacetime 4-vector 0xyz) =(ω,ckx,cky,ckz) transformation

Lecture 29.  : Relativistic wave mechanics VII. Space-time geometry and Applications III. (4/26/2016)   View on YouTube

(Unit 3 p.28-42)


Review of hyper-trigonometry ( tanhρ, sinhρ, and coshρ, sechρ, and cschρ, cothρ )

and co-trigonometry ( sinσ, tanσ, and secσ, cosσ, and cotσ, cscσ )

Review of “Occam-sword” geometry and wave parameters for phase and group motion

Wave parameter symmetry points


Yet another view: Epstein’s space-proper-time approach to SR and stellar aberration k-angle σ

Review of proper time relations and basis of Epstein’s cosmic speedometer

Epstein vs Einstein-Minkowski geometry of relativity

     Einstein time dilation

          Lorentz space contraction

               Time-simultaneity-breaking

                         Velocity addition


Twin-paradox resolution in space-proper-time


Spectral details of per-spacetime 4-vector 0, ωx, ωy, ωz) =(ω, ckx, cky, ckz) transformation

Lecture 30.  : Relativistic quantum mechanics I. Basic theory and Applications IV. (4/28/2016)   View on YouTube

(Unit 3 p.45-61)


Using (some) wave parameters to develop relativistic quantum theory

Low rapidity approximations to υphase and cκphase match to Newtonian KE and momentum

How Mc2 pops right up

Exact υphase gives exact Planck-Einstein energy formulas (1900-1905)

Exact cκphase gives exact Bohr momentum and dispersion formulas (1921-1927)

Bohr-Schrodinger approximation to dispersion (Who threw away the Mc2 ?!!)


“What’s the Matter with Mass?” Definition(s) of relativistic and quantum mechanical mass

(1) Einsteinian rest mass   (2) Galilean momentum mass   (3) Newtonian effective mass

Three Faces of Eve: A photon’s split-mass personality


Relativistic action S and Lagrangian-Hamiltonian relations: How invariant phase works

The Legendre transformation relations

Deriving Lagrangian and Hamiltonian functions

Geometry of 1st Lagrangian and 1st Hamiltonian equations

Poincare invariant action differential

Hamilton-Jacobi equations

     How Hamilton-Jacobi derives Schrodinger-op equations

          How Huygens contact transformations determine motion

Lecture 31.  : Dynamics - Quantizing wave variables of phase and amplitude (5/3/2016)   View on YouTube

(Unit 3 p.45-64)


Review of wave parameters used to develop relativistic quantum theory

Bohr-Schrodinger (BS) approximation throws out Mc2 (Is frequency really relative?)

Effect on group velocity (None) and phase velocity (Absurd)


1st Quantization: Quantizing phase variables km and ω(km)

Understanding how quantum dynamics and transitions involve “mixed” states

Square well example of mixing unequal frequencies

Circle well or ring example of mixing equal or unequal frequencies


Mixing unequal amplitudes makes “Galloping” wave: Analogy of (SWR, SWQ) to (Vgroup, Vphase)

Analogy with optical polarization geometry and Kepler orbits

Super-luminal speed and Feynman-Wheeler pair-creation switchbacks


2nd Quantization: Quantizing wave amplitudes AN and invariance of photon number

Analogy 1: Many CW (Continuous Waves) add up to make PW (Pulse Waves)

Analogy 2: Many Photon-Number-Modes add up to make Coherent-Laser-Modes

Heisenberg ΔυΔt~1~ΔκΔx analogous to ΔNΔphase~1 uncertainty relations


Electromagnetic wave mode energy: Maxwell vs. Planck-Einstein

1st quantization for wave phase variables and classical energy of E, B , and A fields

2nd quantization for wave and Planck quantum energy of E, B , and A fields

Scaling E-waves to mime quantum Ψ-waves and ψ-waves


Relativistic effects on charge, current, and Maxwell Fields

Lecture 32.  : Dynamics II. - Spectroscopy, transitions, and acceleration (5/5/2016)   View on YouTube

(Unit 3 p.45-61 - 4.26.16)


Relativity relates charge, current, and magnetic fields

Geometric derivation of magnetic constant μ0 from electric ε0


Lorentz-Poincare symmetry and energy-momentum spectral conservation rules

Review of 2nd-quantization “photon” number N and 1st-quantization wavenumber κ=m

Sketches of atomic and molecular spectroscopy


Relativistic optical transitions and Compton recoil formulae

Feynman diagram geometry

Recoils shifts

     Compton recoil related to rocket velocity formula

          Geometric transition coordinate grids


Relawavity in accelerated frames

Relawavity in accelerated frames

Analysis of constant-g grid compared to zero-g Minkowski frame

     Animation of mechanics and metrology of constant-g grid