Lecture 4.  Kinetic Derivation of 1D Potentials and Force Fields (9/3/2015)   

(Ch. 6, and Ch. 7 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 m₂ keeps its action

An interesting wave analogy: The “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 5.  Dynamics of Potentials and Force Fields (9/8/2015)   

(Ch. 7 and part of Ch. 8 of Unit 1)

(From Lect 5.) A lesson in geometry of fractions and fractals: Ford Circles and Farey Sums

Link → ["Fractions" - Lester. R. Ford, Am. Math. Monthly 45,586(1938)]
["On a Curious Property of Vulgar Fractions" - John Farey, Phil. Mag. 47, 385 (1816)]

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 6.  Geometry of common power-law potentials I. (9/10/2015)   

(Ch. 9 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

Lecture 7.  Geometry of common power-law potentials II. (9/15/2015)   

(Ch. 9 of Unit 1)

Review of “Sophomore-physics Earth” field geometry

“Outside” Coulomb geometry of -kr^-1-potential and -kr^-2-force field

“Inside” Oscillator geometry of kr²/2 potential and -kr¹ force field

Easy-to-remember geo-solar constants

Geometry and algebra of idealized “Sophomore-physics Earth” fields

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

Contact-geometry of potential curve(s) and “kite” geometry

“Ordinary-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”

Isotropic Harmonic Oscillator phase dynamics in uniform-body orbits

Dual phasor construction of elliptic oscillator orbits

Integrating IHO equations by phasor geometry

Link → BoxIt - Simulation with IHO Potential at http://www.uark.edu/ua/modphys/markup/BoxItWeb.html
Link → RelaWavity - IHO Elliptical orbits at http://www.uark.edu/ua/modphys/markup/RelaWavityWeb.html?plotType=1|0
Link → RelaWavity - IHO Ellipse Exegesis at http://www.uark.edu/ua/modphys/markup/RelaWavityWeb.html?plotType=1|1

Link → CoulIt - Simulation with Coulomb Potential at http://www.uark.edu/ua/modphys/markup/CoulItWeb.html

Lecture 8.  Kepler Geometry of IHO (Isotropic Harmonic Oscillator) Elliptical Orbits (9/17/2015)   

(Ch. 9 and Ch. 11 of Unit 1)

Review of IHO orbital phasor “clock” dynamics in uniform-body with two “movie” examples

Constructing 2D IHO orbits using Kepler anomaly plots

Mean-anomaly and eccentric-anomaly geometry

Calculus and vector geometry of IHO orbits

A confusing introduction to Coriolis-centrifugal force geometry

Some Kepler’s “laws” for all central (isotropic) force F(r) 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

Introduction to dual matrix operator contact geometry (based on IHO orbits)

Quadratic form ellipse r•Q•r=1 vs.inverse form ellipse p•Q^-1•p=1

Duality norm relations (r•p=1)

Q-Ellipse tangents r′ normal to dual Q^-1-ellipse position p (r′•p=0=r•p′)

Operator geometric sequences and eigenvectors

Alternative scaling of matrix operator geometry

Vector calculus of tensor operation

Q: Where is this headed?   A: Lagrangian-Hamiltonian duality

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 9.  Quadratic form geometry and Development of mechanics of Lagrange and Hamilton (9/22/2015)   

(Ch. 12 of Unit 1 and Ch. 4-5 of Unit 7)

Review of partial differential calculus

Chain rule and order ∂²Ψ/∂x∂y = ∂²Ψ/∂y∂x symmetry

Scaling transformation between Lagrangian and Hamiltonian views of KE

Introducing 0^th Lagrange and 0^th Hamilton differential equations of mechanics

Introducing 1^st Lagrange and 1^st Hamilton differential equations of mechanics

Introducing the Poincare´ and Legendre contact transformations

Geometry of Legendre contact transformation (Preview of Unit 8 relativistic quantum mechanics)

Example from thermodynamics

Legendre transform: special case of General Contact Transformation (Lights, Camera, ... ACTION!)

An elementary contact transformation from sophomore physics

Algebra-calculus development of “The Volcanoes of Io” and “The Atoms of NIST”

Intuitive-geometric development of ” ” ” and ” ” ”

Link → CoulIt - Simulation of the Volcanoes of Io
Link → RelaWavity - Physical Terms H(p) & L(u)

Lecture 10.  Equations of Lagrange and Hamilton mechanics in Generalized Curvilinear Coordinates (GCC) (9/24/2015)   

(Ch. 12 of Unit 1, Ch. 1-5 of Unit 2, and Ch. 1-5 of Unit 3)

Quick Review of Lagrange Relations in Lectures 8-9

Using differential chain-rules for coordinate transformations

Polar coordinate example of Generalized Curvilinear Coordinates (GCC)

Getting the GCC ready for mechanics: Generalized velocity and Jacobian Lemma 1

Getting the GCC ready for mechanics: Generalized acceleration and Lemma 2

How to say Newton’s “F=ma” in Generalized Curvilinear Coords.

Use Cartesian KE quadratic form KE=T=v•M•v/2 and F=M•a to get GCC force

Lagrange GCC trickery gives Lagrange force equations

Lagrange GCC trickery gives Lagrange potential equations (Lagrange 1 and 2)

GCC Cells, base vectors, and metric tensors

Polar coordinate examples: Covariant E_m vs. Contravariant E^m

Covariant metric g_mn vs. Invariant δ_mⁿ vs. Contravariant metric g^mn

Lagrange prefers Covariant g_mn with Contravariant velocity

GCC Lagrangian definition

GCC “canonical” momentum p_m definition

GCC “canonical” force F_m definition/p>

Coriolis “fictitious” forces (… and weather effects)

Lecture 11.  Hamiltonian vs. Lagrange mechanics in Generalized Curvilinear Coordinates (GCC) (10/6/2015)   

(Unit 1 Ch. 12, Unit 2 Ch. 2-7, Unit 3 Ch. 1-3)

Review of Lectures 9-12 procedures:

Lagrange prefers Covariant metric g_mn with Contravariant velocity

Hamilton prefers Contravariant metric g^mn with Covariant momentum p_m

Deriving Hamilton’s equations from Lagrange’s equations

Expressing Hamiltonian H(momentum p_m,qⁿ) using g^mn and covariant momentum p_m

Polar-coordinate example of Hamilton’s equations

     Hamilton’s equations in Runga-Kutta (computer solution) form

Examples of Hamiltonian mechanics in effective potentials

I_sotropic H_armonic O_scillator in polar coordinates and effective potential (Old Mac OS & Web {In developement} simulations)

Coulomb orbits in polar coordinates and effective potential (Old Mac OS)

Parabolic and 2D-IHO orbital envelopes

Clues for future assignment _ (Web Simulation: CouIIt)

Examples of Hamiltonian mechanics in phase plots

1D Pendulum and phase plot (Web Simulations: Pendulum, Cycloidulum (Constrained Pendulum), and JerkIt (Vertically Driven Pendulum))

1D-HO phase-space control (Old Mac OS and Web Simulation of “Catcher in the Eye”)

Links to simulations embedded above

Lecture 12.  Poincare, Lagrange, Hamiltonian, and Jacobi mechanics (10/8/2015)   

(Unit 1 Ch. 12, Unit 2 Ch. 2-7, Unit 3 Ch. 1-3, Unit 7 Ch. 1-2)

Parabolic and 2D-IHO orbital envelopes ( Review of Lecture 9 p.56-81 and a generalization.)

Clues for future assignment (Web Simulation: CouIIt)

Examples of Hamiltonian mechanics in phase plots

1D Pendulum and phase plot (Web Simulations: Pendulum, Cycloidulum (Constrained Pendulum), and JerkIt (Vertically Driven Pendulum))

1D-HO phase-space control (Old Mac OS and Web Simulation of “Catcher in the Eye”)

Exploring phase space and Lagrangian mechanics more deeply

A weird “derivation” of Lagrange’s equations

Poincare identity and Action, Jacobi-Hamilton equations

How Classicists might have “derived” quantum equations

Huygen’s contact transformations enforce minimum action

How to do quantum mechanics if you only know classical mechanics

(“Color-Quantization” simulations: Davis-Heller “Color-Quantization” or “Classical Chromodynamics”)

Lecture 13.  Complex Variables, Series, and Field Coordinates I. (10/13/2015)   

(Ch. 10 of Unit 1)

Lecture 14.  Complex Variables, Series, and Field Coordinates II. (10/15/2015)   

(Ch. 10 of Unit 1)

Lecture 15.  Introducing GCC Lagrangian`a la Trebuchet Dynamics (10/22/2015)   

Ch. 1-3 of Unit 2 and Unit 3 (Mostly Unit 2.)

The trebuchet (or ingenium) and its cultural relevancy (3000 BCE to 21^st See Sci. Am. 273, 66 (July 1995))

The medieval ingenium (9^th to 14^th century) and modern re-enactments

Human kinesthetics and sports kinesiology

Review of Lagrangian equation derivation from Lecture 10 (Now with trebuchet model)

Coordinate geometry, Jacobian, velocity, kinetic energy, and dynamic metric tensor γ_mn

Structure of dynamic metric tensor γ_mn

Basic force, work, and acceleration

Lagrangian force equation

Canonical momentum and γ_mn tensor

Summary of Lagrange equations and force analysis (Mostly Unit 2.)

Forces: total, genuine, potential, and/or fictitious

Geometric and topological properties of GCC transformations (Mostly from Unit 3.)

Multivalued functionality and connections

Covariant and contravariant relations

Tangent space vs. Normal space

Metric g_mn tensor geometric relations to length, area, and volume

Lecture 16.  GCC Lagrange and Riemann Equations for Trebuchet (10/27/2015)   

(Ch. 1-5 of Unit 2 and Unit 3)

Review (Mostly Unit 2.): Was the Trebuchet a dream problem for Galileo? Not likely.

Forces in Lagrange force equation: total, genuine, potential, and/or fictitious

Geometric and topological properties of GCC transformations (Mostly from Unit 3.)

Trebuchet Cartesian projectile coordinates are double-valued

Toroidal “rolled-up” (q¹=θ, q²=φ)-manifold and “Flat” (x=θ, y=φ)-graph

Review of covariant E_n and contravariant E^m vectors: Jacobian J vs. Kajobian K

Covariant metric g_mn vs. contravariant metric g^mn (Lect. 10 p.43)

Tangent {E_n}space vs. Normal {E^m}space

Covariant vs. contravariant coordinate transformations

Metric g_mn tensor geometric relations to length, area, and volume

Lagrange force equation analysis of trebuchet model (Mostly from Unit 2.)

Review of trebuchet canonical (covariant) momentum and mass metric γ_mn (Lect. 15 p. 77)

Review and application of trebuchet covariant forces F_θ and F_φ (Lect. 15 p. 69)

Riemann equation derivation for trebuchet model

Riemann equation force analysis

2nd-guessing Riemann equation?

Lecture 17.  Hamilton Equations for Trebuchet and Other Things (10/29/2015)   

(Ch. 5-9 of Unit 2)

Review of Hamiltonian equation derivation (Elementary trebuchet)

Hamiltonian definition from Lagrangian and γ_mn tensor

Hamilton’s equations and Poincare invariant relations

Hamiltonian expression and contravariant γ^mn tensor

Hamiltonian energy and momentum conservation and symmetry coordinates

Coordinate transformation helps reduce symmetric Hamiltonian

Free-space trebuchet kinematics by symmetry

Algebraic approach

Direct approach and Superball analogy

Trebuchet vs Flinger and sports kinematics

Many approaches to Mechanics

Lecture 18.  Reimann-Christoffel equations and covariant derivative (11/3/2015)   

(Ch. 5-9 of Unit 2)

Covariant derivative and Christoffel Coefficients Γ_ij;k and Γ_ij;^k

Christoffel g-derivative formula

What’s a tensor? What’s not?

General Riemann equations of motion (No explicit t-dependence and fixed GCC)

Riemann-forms in cylindrical polar OCC (q¹ = ρ, q² = φ, q³ = z)

Christoffel relation to Coriolis coefficients

Mechanics of ideal fluid vortex

Separation of GCC Equations: Effective Potentials

Small (n_ρ:m_φ)-periodic and quasi-periodic oscillations

2D Spherical pendulum“Bowl-Bowling” and the “I-Ball”

     (n_ρ:m_φ)=(2:1) vs (1:1) periodic and quasi-periodic orbits

Cycloidal ruler&compass geometry

(To be applied to mechanics in electromagnetic fields and collisional rotation in following lectures.)

Lecture 19.  Electromagnetic Lagrangian and charge-field mechanics (11/5/2015)   

(Ch. 2.8 of Unit 2)

Charge mechanics in electromagnetic fields Vector analysis for particle-in-(A,Φ)-potential Lagrangian for particle-in-(A,Φ)-potential Hamiltonian for particle-in-(A,Φ)-potential Canonical momentum in (A,Φ) potential Hamiltonian formulation Hamilton’s equations Crossed E and B field mechanics Classical Hall-effect and cyclotron orbit orbit equations Vector theory vs. complex variable theory Mechanical analog of cyclotron and FBI rule Cycloidal ruler&compass geometry Cycloidal geometry of flying levers      Practical poolhall application

This mechanical analog of (Ex,Bz) field mimics A-field with tabletop v-field

Lecture 20.  Introduction to classical oscillation and resonance (11/10/2015)   

(Ch. 1 of Unit 4)

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 21.  Introduction to coupled oscillation and eigenmodes (11/12/2015)   

(Ch. 2-4 of Unit 4 11.12.15)

2D harmonic oscillator equations Lagrangian and matrix forms and Reciprocity symmetry 2D harmonic oscillator equation eigensolutions Geometric method Matrix-algebraic eigensolutions with example M = 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

2D-HO eigensolution example with bilateral (B-Type) symmetry

Mixed mode beat dynamics and fixed π/2 phase

2D-HO eigensolution example with asymmetric (A-Type) symmetry

Initial state projection, mixed mode beat dynamics with variable phase

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

Hamilton-Pauli spinor symmetry (ABCD-Types)

Lecture 22.  Introduction to Spinor-Vector resonance dynamics (11/17/2015)   

(Ch. 2-4 of Unit 4 Ch. 6-7 of Unit 6)

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: ∂²_tx=-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^-iσµ^ωµ^t

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^-iσµ^ωµ^t

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 S_A=S_Z, Balance S_B =S_X, and Chirality S_C =S_Y

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 23.  U(2)~R(3) algebra/geometry in classical or quantum theory (11/19/2015)   Part 1  Part 2

(Classical Mechanics with a BANG! Units 4-6, Quantum Theory for Computer Age - Ch. 10A-B of Unit 3)

(Principles of Symmetry, Dynamics, and Spectroscopy - Sec. 1-3 of Ch. 5 and Ch. 7)

Reviewing fundamental Euler R(αβγ) and Darboux R[ϕϑΘ] representations of U(2) and R(3)

Euler-defined state |αβγ〉 described by Stoke’s S-vector, phasors, or ellipsometry

Darboux defined Hamiltonian H = [ϕϑΘ] = exp(-iΩ•S)·t and angular velocity Ω(φθ)·t=Θ-vector

Euler-defined operator R(αβγ) derived from Darboux-defined R[ϕϑΘ] and vice versa

Euler R(αβγ) rotation Θ =0-4π-sequence [ϕϑ] fixed (and “real-world” applications)

Quick U(2) way to find eigen-solutions for 2-by-2 Hamiltonian H

The ABC’s of U(2) dynamics-Archetypes

Asymmetric-Diagonal A-Type motion

Bilateral-Balanced B-Type motion

Circular-Coriolis… C-Type motion

The ABC’s of U(2) dynamics-Mixed modes

AB-Type motion and Wigner’s Avoided-Symmetry-Crossings

ABC-Type elliptical polarized motion

Ellipsometry using U(2) symmetry coordinates

Conventional amp-phase ellipse coordinates

Euler Angle (αβγ) ellipse coordinates

Addenda: U(2) density matrix formalism

Bloch equation for density operator

Lecture 24.  Parametric Resonance and Multi-particle Wave Modes (11/24/2015)  

(Ch. 7-8 of Unit 4 11.24.15)

Two Kinds of Resonance: Linear-additive vs. Nonlinear-multiplicative (Parametric resonance)

Coupled rotation and translation (Throwing revisited: trebuchet, atlatl, etc.)

Schrodinger wave equation related to Parametric resonance dynamics

Electronic band theory and analogous mechanics

Wave resonance in cyclic symmetry

Harmonic oscillator with cyclic C₂ symmetry

C₂ symmetric (B-type) modes

Harmonic oscillator with cyclic C₃ symmetry

C₃ symmetric spectral decomposition by 3rd roots of unity

Resolving C₃ projectors and moving wave modes

Dispersion functions and standing waves

C₆ symmetric mode model:Distant neighbor coupling

C₆ spectra of gauge splitting by C-type symmetry(complex, chiral, coriolis, current, ...)

C_N symmetric mode models: Made-to order dispersion functions

Quadratic dispersion models: Super-beats and fractional revivals

Phase arithmetic

Algebra and geometry of resonant revivals: Farey Sums and Ford Circles

Relating C_N symmetric H and K matrices to differential wave operators

Lecture 25.  Introduction to Orbital Dynamics (12/1/2015)   

(Ch. 2-4 of Unit 5 12.01.15)

Orbits in Isotropic Harmonic Oscillator and Coulomb Potentials

Effective potentials for IHO and Coulomb orbits     Review: “3steps from Hell” (Lect. 7 Ch. 9 Unit 1)

Stable equilibrium radii and radial/angular frequency ratios

Classical turning radii and apogee/perigee parameters     ← (A mysterious similarity appears)

Polar coordinate differential equations     ← (A mysterious similarity appears)

Quadrature integration techniques

     Detailed orbital functions     ← (A mysterious similarity appears)

Relating orbital energy-momentum to conic-sectional orbital geometry

     Kepler equation of time and phase geometry

Geometry and Symmetry of Coulomb orbits

Detailed elliptic geometry

Detailed hyperbolic geometry

Lecture 26.  Geometry and Symmetry of Coulomb Orbital Dynamics I. (12/3/2015)   

(Ch. 2-4 of Unit 5 12.03.15)

Rutherford scattering and hyperbolic orbit geometry

Backward vs forward scattering angles and orbit construction example

Parabolic “kite” and orbital envelope geometry

Differential and total scattering cross-sections

Eccentricity vector ε and (ε,λ)-geometry of orbital mechanics

Projection ε•r geometry of ε-vector and orbital radius r

Review and connection to usual orbital algebra (previous lecture)

Projection ε•p geometry of ε-vector and momentum p=mv

General geometric orbit construction using ε-vector and (γ,R)-parameters

Derivation of ε-construction by analytic geometry

Coulomb orbit algebra of ε-vector and Kepler dynamics of momentum p=mv

Example of complete (r,p)-geometry of elliptical orbit

Connection formulas for (a,b) and (ε,λ) with (γ,R)

Lecture 27.  Geometry and Symmetry of Coulomb Orbital Dynamics II. (12/4/2015)   

(Ch. 2-4 of Unit 5 12.05.15)

Review of Eccentricity vector ε and (ε,λ)-geometry of orbital mechanics    ← Review of lecture 26

Analytic geometry derivation of ε-construction    ← Review of lecture 26

Connection formulas for (a,b) and (ε,λ) with (γ,R)    ← Review of lecture 26

Detailed ruler & compass construction of ε-vector and orbits

(R = -0.375 elliptic orbit)

(R = +0.5 hyperbolic orbit)

Properties of Coulomb trajectory families and envelopes

Graphical ε-development of orbits

Launch angle fixed-Varied launch energy

Launch energy fixed-Varied launch angle

Launch optimization and orbit family envelopes

Lecture 28.  Multi-particle and Rotational Dynamics (12/7/2015)  < 

(Ch. 2-7 of Unit 6 12.12.14)

2-Particle orbits

Ptolemetric or LAB view and reduced mass

Copernican or COM view and reduced coupling

2-Particle orbits and scattering: LAB-vs.-COM frame views

Ruler & compass construction (or not)

Rotational equivalent of Newton’s F=dp/dt equations: N=dL/dt

How to make my boomerang come back

The gyrocompass and mechanical spin analogy

Rotational momentum and velocity tensor relations

Quadratic form geometry and duality (again)

Angular velocity ω-ellipsoid vs. angular momentum L-ellipsoid

Lagrangian ω-equations vs. Hamiltonian momentum L-equation

Rotational Energy Surfaces (RES) and Constant Energy Surfaces (CES)

Symmetric, asymmetric, and spherical-top dynamics (Constant L)

BOD-frame cone rolling on LAB frame cone

Deformable spherical rotor RES and semi-classical rotational states and spectra

Cycloidal geometry of flying levers

     Practical poolhall application

Lecture 29.  Classical Constraints: Comparing various methods (12/8/2015)   

(Ch. 9 of Unit 3)

Some Ways to do constraint analysis

Way 1. Simple constraint insertion

Way 2. GCC constraint webs

Find covariant force equations

Compare covariant vs. contravariant forces

Other Ways to do constraint analysis

Way 3. OCC constraint webs

Sketch of atomic-Stark orbit parabolic OCC analysis

     Classical Hamiltonian separability

Way 4. Lagrange multipliers

Lagrange multiplier as eigenvalues

Multiple multipliers

“Non-Holonomic” multipliers

Cycloid-like curves for rolling constraints

Lecture 30.  RelaWavity and a novel introduction to relativistic mechanics I. (12/10/2015)   

(Ch. 6 of Unit 8 12.10.15)

How Doppler shifts of cavity waves exhibit relativistic Lorentz transformations

Optical interference “baseball-diamond” displays phase and group velocity

Doppler shifted “baseball-diamond” displays Lorentz frame transformation

Developing Axioms to update Galileo’s relativity: A critical look at c-axioms

Einstein’s PW (Pulse-Wave) Axiom

Evenson’s CW (Continuous Wave) Axiom and Occam’s Razor

Analyzing wave velocity by per-space-per-time and space-time graphs

Introducing optical Doppler effects

Clarifying Evenson’s CW Axiom using Doppler effects

Galileo’s Revenge (part1): Galilean Doppler-shift arithmetic using rapidity ρ

Developing optical “baseball-diamond” and relativistic ρ-functions and transformations

Details of 1CW wavefunctions and phasors

Details of 2CW wavefunctions in rest frame

Galileo’s Revenge (part2): Galilean addition of phasor angular velocity

Structure of rest frame “baseball-diamonds”

Details of 2CW wavefunctions of moving frame velocities of phase and group waves

16 coefficients of relativistic 2CW interference

Two “famous-name” coefficients and the Lorentz transformation

Thales geometry of Lorentz transformation

Lecture 31.  RelaWavity and a novel introduction to relativistic mechanics II. (12/17/2015)   

(Ch. 6-8 of Unit 8 12.10.15)

Review of 16 functions of ρ and related geometric approach to relativity

Animation of e^ρ=2 spacetime and per-spacetime plots

Rapidity ρ related to stellar aberration angle σ and L. C. Epstein’s approach to relativity

Longitudinal hyperbolic ρ-geometry connects to transverse circular σ-geometry

“Occams Sword” and summary of 16 parameter functions of ρ and σ

Applications to optical waveguide, spherical waves, accelerator radiation

     Learning about sin! and cos and...

Derivation of relativistic quantum mechanics

What’s the matter with mass? Shining some light on the Elephant in the room

Relativistic action and Lagrangian-Hamiltonian relations

Poincare’ and Hamilton-Jacobi equations

Relativistic optical transitions and Compton recoil formulae

Feynman diagram geometry

Compton recoil related to rocket velocity formula

Relation of 2^nd quantization amplitude “photon” N and 1^st quantization wavenumber κ

in accelerated frames

Laser up-tuning by Alice and down-tuning by Carla makes g-acceleration grid

Analysis of constant-g grid compared to zero-g Minkowsi grid

Animation of mechanics and metrology of constant-g grid

Link →   DAMOP Annual Meeting RelaWavity Web Portal - OSU Columbus, OH 6/10/2015