 2014 Detailed List of Lecture Topics

# Lecture 1. Axiomatic development of classical mechanics (8.26.14) Part 1 Part 2 Part 3 (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 (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

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

# Lecture 2. Analysis of 1D 2-Body Collisions I. (8.28.14) Part 1 Part 2 (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 (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)

Example of (V1,V2) and (y1,y2) data for high mass ratios: m1/m2=49, 100,...

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

# Lecture 3. Analysis of 1D 2-Body Collisions II. (9.02.14) Lecture 4. Analysis of 1D 2-Body Collisions III.(9.04.14)

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

Review of (V1,V2) and (y1,y2) geometry and X2 launcher in box

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)

Example of (V1,V2) and (y1,y2) data for high mass ratios: m1/m2=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 = CM

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 for m1/m2=3)

Solutions to Exercises 1.4.1 and 1.4.2

# Lecture 5. Kinetic Derivation of 1D Potentials and Force Fields (9.09.14) Part 1 Part 2 (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

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)]
Link → Quantum Revivals of Morse Oscillators and Farey-Ford Geometry - 2013

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)]

# Lecture 6. Dynamics of Potentials and Force Fields I. (9.11.14)

(Ch. 7 and 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

Examples: (a) Constant force (like kidee pool) (b) Linear force (like balloon)

Some physics of dare-devil-divers

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

Geometry and dynamics of 2-ball bounce (again with feeling)

The parable of RumpCo. vs CrapCorp.

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

Geometry and dynamics of 3-ball bounce

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

Other bangings-on: The western buckboard and Newton’s balls

Crunch energy geometry of freeway crashes and related things

Crunch energy played backwards: This really is “Rocket-Science”

A Thales construction for momentum-energy

# Lecture 7. Dynamics of Potentials and Force Fields II. (9.16.14) (Ch. 7 and Ch. 8 of Unit 1)

Potential energy geometry of Superballs and related things

Thales geometry and “Sagittal approximation” to force law

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 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

(Leads to Sagittal potential analysis of 2, 3, and 4 body towers)

Many-body 1D collisions

Elastic examples: Western buckboard

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

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

# Lecture 8. Geometry of common power-law potentials (9.18.14) 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 9. Kepler Geometry of IHO (Isotropic Harmonic Oscillator) Elliptical Orbits (9.23.14) (Ch. 9 and Ch. 11 of Unit 1)

Constructing 2D IHO orbits by phasor plots

Phasor “clock” geometry

Integrating IHO equations by phasor geometry

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 (Derived rigorously later in Ch. 12)

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

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

Angular momentum invariance of Coulomb: F(r)=-GMm/r2 with U(r)=-GMm·/r (Derived later in Unit 5)

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

Total energy E=KE+PE invariance of Coulomb: F(r)=-GMm/r2 (Derived later in Unit 5)

Brief introduction to matrix quadratic form geometry

# Lecture 10. Geometry of Dual Quadratic Forms: Lagrange vs Hamilton (9.24.14) (Ch. 11 and Ch. 12 of Unit 1)

Review of dual IHO elliptic orbits (Lecture 9)

Construction by Phasor-pair projection (with example of dual-ellipses)

Construction by Kepler anomaly projection

Introduction to dual matrix operator geometry

Quadratic form ellipse rQr=1 vs. inverse form ellipse pQ-1p=1

Duality norm relations (rp=1)

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

Operator geometric sequences and eigenvectors

Alternative scaling of matrix operator geometry

Vector calculus of tensor operation

Introduction to Lagrangian-Hamiltonian duality

Review of partial differential relations

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

Duality relations of Lagrangian and Hamiltonian ellipse

Introducing the 1st (partial) differential equations of mechanics

# Lecture 11. Quadratic form geometry and development of mechanics of Lagrange and Hamilton (9.30.14) (Ch. 12 of Unit 1 and Ch. 4-5 of Unit 7)

Scaling transformation between Lagrangian and Hamiltonian views of KE (Review of Lecture 10)

Introducing 1st Lagrange and Hamilton differential equations of mechanics (Review Of Lecture 10)

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!)

A general contact transformation from sophomore physics

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

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

# Lecture 12. Equations of Lagrange and Hamilton mechanics in Generalized Curvilinear Coordinates (GCC) (10.2.14) (Ch. 12 of Unit 1 and Ch. 1-5 of Unit 2 and Ch. 1-5 of Unit 3)

Quick Review of Lagrange Relations in Lectures 9-11

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 Em vs. Contravariant Em

Covariant metric gmn vs. Invariant δmn vs. Contravariant metric gmn

Lagrange prefers Covariant gmn with Contravariant velocity

GCC Lagrangian definition

GCC “canonical” momentum pm definition

GCC “canonical” force Fm definition/p>

Coriolis “fictitious” forces (… and weather effects)

# Lecture 13. Hamiltonian vs. Lagrange mechanics in Generalized Curvilinear Coordinates (GCC) (10.7.14) (Unit 1 Ch. 12, Unit 2 Ch. 2-7, Unit 3 Ch. 1-3)

Review of Lectures 9-12 procedures:

Lagrange prefers Covariant metric gmn with Contravariant velocity Hamilton prefers Contravariant metric gmn with Covariant momentum pm

Deriving Hamilton’s equations from Lagrange’s equations

Expressing Hamiltonian H(momentum pm,qn) using gmn and covariant momentum pm

Polar-coordinate example of Hamilton’s equations

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

Examples of Hamiltonian mechanics in effective potentials

Isotropic Harmonic Oscillator in polar coordinates and effective potential (Simulation)

Coulomb orbits in polar coordinates and effective potential (Simulation)

Parabolic and 2D-IHO orbital envelopes

Clues for future assignment _ (Simulation)

Examples of Hamiltonian mechanics in phase plots

1D Pendulum and phase plot (Simulation)

1D-HO phase-space control (Simulation)

# Lecture 14. Poincare, Lagrange, Hamiltonian, and Jacobi Mechanics (10.9.14) (Unit 1 Ch. 12, Unit 2 Ch. 2-7, Unit 3 Ch. 1-3, Unit 7 Ch. 1-2)

Review of Lecture 12 relations

Examples of Hamiltonian mechanics in phase plots

1D Pendulum and phase plot (Simulation)

1D-HO phase-space control (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 15. Complex Variables, Series, and Field Coordinates I. (10.14.14) (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 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 16. Complex Variables, Series, and Field Coordinates II. (10.16.14) (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. 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.

# Lecture 17. Introducing GCC Lagrangian`a la Trebuchet Dynamics (10.23.14) (Ch. 1-3 of Unit 2 and Unit 3)

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

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

Human kinesthetics and sports kinesiology

Cartesian to GCC transformations (Mostly Unit 2.)

Jacobian relations

Kinetic energy calculation

Dynamic metric tensor γmn

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

Multivalued functionality and connections

Covariant and contravariant relations

Metric tensors

# Lecture 18. GCC Lagrange and Riemann Equations for Trebuchet or “How do we ignore all those constraint forces?” (10.23.14 & 10.28.14) (Ch. 1-5 of Unit 2 and Unit 3)

Review of Lagrangian equation derivation (Elementary trebuchet) (Mostly Unit 2.)

Coordinate geometry, kinetic energy, and dynamic metric tensor γmn

Force, work, and acceleration

Lagrangian force equation

Canonical momentum and γmn tensor

Equations of motion and force analysis (Mostly Unit 2.)

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

Lagrange equation forms

Riemann equation forms

2nd-guessing Riemann? (More like Unit 3.)

# Lecture 19. Hamilton Equations for Trebuchet and Other Things (10.28.14)  Lects. 19 & 20 (Ch. 5-9 of Unit 2)

Review of Hamiltonian equation derivation (Elementary trebuchet)

Hamiltonian definition from Lagrangian and γmn tensor

Hamilton’s equations and Poincare invariant relations

Hamiltonian expression and contravariant γmn 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 20. Reimann-Christoffel equations and covariant derivative (10.30.14)

(Ch. 4-7 of Unit 3)

Covariant derivative and Christoffel Coefficients Γij;k and Γij;k

Christoffel g-derivative formula

What’s a tensor? What’s not?

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

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

Separation of GCC Equations: Effective Potentials

Cycloid vs Pendulum

# Lecture 21. Electromagnetic Lagrangian and charge-field mechanics (11.4.14 - 11.6.14) (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

Crossed E and B field mechanics

Classical Hall-effect and cyclotron orbits

Vector theory vs. complex variable theory

Mechanical analog of cyclotron and FBI rule

# Lecture 22. Reimann-Christoffel equations and covariant derivative (11.6.14) (Ch. 2.8 of Unit 2)

Separation of GCC Equations: Effective Potentials

2D Spherical pendulum or “Bowl-Bowling”

Cycloidal ruler & compass geometry

Cycloid as brachistichrone with various geometries

Cycloid as tautochrone

Cycloidulum vs Pendulum

Cycloidal geometry of flying levers

Practical poolhall application

# Lecture 23. Classical Constraints: Comparing various methods (11.11.14) (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

Preview of atomic-Stark orbits

Classical Hamiltonian separability

Way 4. Lagrange multipliers

Lagrange multiplier as eigenvalues

Multiple multipliers

“Non-Holonomic” multipliers

# Lecture 24. Introduction to classical oscillation and resonance (11.11.14) (Ch. 1 of Unit 4 11.14.13)

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 SolutionGreen’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)

# Lecture 25. Introduction to coupled oscillation and eigenmodes (11/18/2014 & 11/20/2014)

(Ch. 2-4 of Unit 4 11.19.13)

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

Beat, lifetimes, and quality factor q = ω0/2Γ and Q = υ0/2Γ = q/2π

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

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

2D harmonic oscillator equations

Lagrangian and matrix forms

Reciprocity symmetry

2D harmonic oscillator equation eigensolutions

Geometric method

Matrix-algebraic method with example M = Secular eq., Hamilton-Cayley eq., Idempotent projectors, (how eigenvalues⇒eigenvectors)

Spectral decomposition and P-operator expansions (how projectors⇒eigensolutions)

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 fluid phase

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

Hamilton-Pauli spinor symmetry (ABCD-Types)

# Lecture 26. Introduction to Spinor-Vector resonance dynamics (11.25.14)

(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: ∂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

# Lecture 27. Parametric Resonance and Multi-particle Wave Modes (12/2/14) (Ch. 7-8 of Unit 4 12.02.14)

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, …)

CN 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 CN symmetric H and K matrices to differential wave operators

# Lecture 28. Introduction to Orbital Dynamics (12/4/14) (Ch. 2-4 of Unit 5 12.04.14)

Review: “3steps from Hell” (Lect. 7 Ch. 9 Unit 1)

Orbits in Isotropic Harmonic Oscillator and Coulomb Potentials

Effective potentials for IHO and Coulomb orbits

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

Polar coordinate differential equations     ← (A mysterious similarity appears)

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 29. Geometry and Symmetry of Coulomb Orbital Dynamics I. (12/10/14) (Ch. 2-4 of Unit 5 12.11.14)

Rutherford scattering and differential scattering cross-sections

Parabolic “kite” and envelope geometry

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

ε-vector and Coulomb r-orbit geometry

Review and connection to standard development

ε-vector and Coulomb p=mv geometry

ε-vector and Coulomb p=mv algebra

Example with elliptical orbit

Analytic geometry derivation of ε-construction

Algebra of ε-construction geometry

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

# Lecture 30. Geometry and Symmetry of Coulomb Orbital Dynamics II. (12/11/14) (Ch. 2-4 of Unit 5)

Eccentricity vector ε and (ε,λ)-geometry of orbital mechanics ← Review of lectures 28 and 29

ε-vector and Coulomb r-orbit geometry ←

Review and connection to standard development ←

ε-vector and Coulomb p=mv geometry ←

Example with elliptical orbit ←

Analytic geometry derivation of ε-construction

Algebra of ε-construction geometry

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

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 31. Multi-particle and Rotational Dynamics (12/12/2014)

(Ch. 2-7 of Unit 6)

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