Text Colors |

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

Geometry of momentum conservation axiom

Totally Inelastic “ka-runch”collisions^{*}

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

Comments on idealization in classical models

Geometry of Galilean translation symmetry

45° shift in (V_{1},V_{2})-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

*
Link to Main Classical Mechanics with a Bang! Web Site
Link → http://www.uark.edu/ua/modphys/markup/CMwBangWeb.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
*

(Ch. 3 to Ch. 5 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 (V_{1},V_{2}) data to space-time plots (y_{1}(t),t) and (y_{2}(t),t) plots

Integration of (V_{1},V_{2}) data to space-space plots (y_{1}, y_{2})

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

*
Link to Main Classical Mechanics with a Bang! Web Site
Link → http://www.uark.edu/ua/modphys/markup/CMwBangWeb.html
*

*
* Launch Superball Collision Simulator
Link → http://www.uark.edu/ua/modphys/markup/BounceItWeb.html
*

(Ch. 3, Ch. 4, and Ch. 5 of Unit 1)

Review of (V_{1},V_{2}) and (y_{1},y_{2}) geometry and X2 launcher in box

Integration of (V_{1},V_{2}) data to space-time plots (y_{1}(t),t) and (y_{2}(t),t) plots

Integration of (V_{1},V_{2}) data to space-space plots (y_{1},y_{2})

Example of (V_{1},V_{2}) and (y_{1},y_{2}) data for high mass ratios: m_{1}/m_{2}=49, 100,...

Multiple collisions calculated by matrix operator products

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

“Mass-bang” matrix M, “Floor-bang” matrix F, “Ceiling-bang” matrix C.

Algebra and Geometry of “ellipse-Rotation” group product: R = C•M

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 D_{6}~C_{6v} (Resulting for m_{1}/m_{2}=3)

Group multiplication and product table

Classical collision paths with D_{6}~C_{6v} (Resulting for m_{1}/m_{2}=3)

*
Solutions to Exercises 1.4.1 and 1.4.2
*

*
* Launch Superball Collision Simulator
Link → http://www.uark.edu/ua/modphys/markup/BounceItWeb.html
*

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

Review of (V_{1},V_{2})→(y_{1},y_{2}) relations
High mass ratio M_{1}/m_{2} = 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./y^{3}

Potential field due to many small bounces

Example of 1D-Adiabatic potential U(y)=const./y^{2}

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 M_{1} ball-wall(s) crushing a poor little m_{2}

How m_{2} 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]

(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=½kx^{2} (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 m_{1}, m_{2}, m_{3},... 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”

(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 kr^{2}/2 potential and -kr^{1} force fields

Coulomb geometry of -1/r-potential and -1/r^{2}-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

(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 kr^{2}/2 potential and -kr^{1} 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
*

(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·r^{2}/2

Angular momentum invariance of Coulomb: F(r)=-GMm/r^{2} 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/r^{2}

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
*

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

Review of partial differential calculus

Chain rule and order ∂^{2}Ψ/∂x∂y = ∂^{2}Ψ/∂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)
*

(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_{m}^{n} 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)

(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},q^{n}) 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
*

(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”)

(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 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. |

(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 4. Complex products provide 2D rotation operations. 5. Complex products provide 2D “dot”(•) and “cross”(x) products. Lecture 16 Thur. 10.16.14 starts here ⤵ 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. |

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

(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” (q^{1}=θ, q^{2}=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?

(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

(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 (q^{1} = ρ, q^{2} = φ, q^{3} = 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.)

(Ch. 2.8 of Unit 2)

Charge mechanics in electromagnetic fields Vector analysis for particle-in-( Lagrangian for particle-in-( Hamiltonian for particle-in-( Canonical momentum in ( 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 (E mimics A-field with tabletop |

(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=ω_{0}/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)

(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}_{t}x=-K•x

Hamilton-Pauli spinor symmetry (ABCD-Types)

(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}_{t}x=-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^{-iσ}µ^{ω}µ^{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^{-iσ}µ^{ω}µ^{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
*

(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(-i**Ω•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

(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 C_{2} symmetry

C_{2} symmetric (B-type) modes

Harmonic oscillator with cyclic C_{3} symmetry

C_{3} symmetric spectral decomposition by 3rd roots of unity

Resolving C_{3} projectors and moving wave modes

Dispersion functions and standing waves

C_{6} symmetric mode model:Distant neighbor coupling

C_{6} 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

(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

(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)

(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

(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**=d**p**/dt equations: **N**=d**L**/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

(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

(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

(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
*