2013 Detailed List of Lecture Topics - Unit 1-6

Lecture 1. Axiomatic development of classical mechanics (Ch. 1 and Ch. 2 of Unit 1)8.27.13

Geometry of momentum conservation axiom

Totally Inelastic "ka-runch"collisions

Perfectly Elastic "ka-bong" and Center Of Momentum (COM) symmetry

Geometry of Galilean translation symmetry

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

Lecture 2. Analysis of 1D 2-Body Collisions (Ch. 3 and Ch. 4 of Unit 1) 8.29.13

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)

Lecture 3. Analysis of 1D 2-Body Collisions (Ch. 3, Ch. 4, and Ch. 5 of Unit 1) 9.03.13

Geometry of X2 launcher bouncing in box (Review)

Example of (V1,V2) and (y1, y2) data for high mass ratios: m 1/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 from m1/m2=3)

Lecture 4. Kinetic Derivation of 1D Potentials and Force Fields (Ch. 6, and Ch. 7 of Unit 1) 9.5.13

Review of (V1,V2)(y1,y2) relations (From Lect. 3) Special mass ratio M1/m2 = 3

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-walls 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: Ford Circles [Lester. R. Ford, Am. Math. Monthly 45,586(1938)] and Farey Sums [John Farey, Phil. Mag.(1816)]

Lecture 5. Dynamics of Potentials and Force Fields (Ch. 7 and Ch. 8 of Unit 1) 9.10.13

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

Lecture 6. Many-body 1D collisions (Ch. 9 and Ch. 11 of Unit 1) (9.12.13)

Elastic examples: Western buckboard, Bouncing column, Newton's cradle
Inelastic examples: "Zig-zag geometry" of freeway crashes
Super-elastic examples: This really is "Rocket-Science"
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

Lecture 7. Geometry and Motion of Isotropic Harmonic Oscillators (Ch. 9 and Ch. 11 of Unit 1) (9.17.13)

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"

Isotropic harmonic oscillator dynamics in 1D, 2D, and 3D

Sinusoidal space-time dynamics derived by geometry

Isotropic harmonic oscillator orbits in 1D and 2D (You get 3D for free!)
Constructing 2D Isotropic harmonic oscillator orbits using phasor plots

Examples with x-y phase lag : αx-y= αx-αy =15°, 30°, and ±75°

Lecture 8. Kepler Geometry of IHO (Isotropic Harm. Osc.) Orbits (Ch. 9 and Ch. 11 of Unit 1)(9.19.13)

Constructing 2D IHO orbits by phasor plots

Review of phasor "clock" geometry (From Lecture 7)
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

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)

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)

Lecture 9. Geometry of Dual Quadratic Forms: Lagrange vs Hamilton (Ch. 11 and Ch. 12 of Unit 1) (9.18.12)

Introduction to dual matrix operator geometry

Review of dual IHO elliptic orbits (Lecture 7-8)

Construction by Phasor-pair projection
Construction by Kepler anomaly projection

Operator geometric sequences and eigenvectors

Rescaled description of matrix operator geometry

Vector calculus of tensor operation

Introduction to Lagrangian-Hamiltonian duality

Review of partial differential relations

Chain rule and order symmetry

Duality relations of Lagrangian and Hamiltonian ellipse

Introducing the 1st (partial ∂?/∂? ) differential equations of mechanics

Lecture 10. Quadratic form geometry and mechanics of Lagrange and Hamilton (Ch. 12 of Unit 1) (9.20.12)

Scaling transformation between Lagrangian and Hamiltonian views of KE (Review of Lecture 9)

Introducing the Poincare´ and Legendre contact transformations

Geometry of Legendre contact transformation 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 11. Equations of Lagrange and Hamilton mechanics in GeneralizedCurvilinear Coordinates (GCC) (Ch. 12 of Unit 1 and Ch. 1-5 of Unit 2 and Ch. 1-5 of Unit 3) (9.20.12)

Using differential chain-rules for coordinate transformations

Polar coordinate example of GCC

GCC for mechanics:Generalized velocity and Jacobian Lemma 1 Generalized acceleration and Lemma 2

How to say Newton's "F=ma" in Generalized Curvilinear Coords.

Use Cartesian KE quadratic form KE=T=1/2v•M•v and F=M•a to get GCC force

Lagrange GCC trickery gives Lagrange force equations

GCC Cells, base vectors, and metric tensors: Polar coordinate examples:

Covariant Emvs.Contravariant Em Covariant gmn vs.Invariant δmn vs.Contravariant gmn

Lagrange prefers Covariant gmn with Contravariant velocity

GCC Lagrangian definition "canonical" momentum pm definition and force Fm definition

Coriolis "fictitious" forces (… and weather effects)

Lecture 12. Hamilton vs. Lagrange mechanics in GCC (Unit 1 Ch. 12, Unit 2, Unit 3)(10.2.12)

Review of Lectures 9-11 procedures:

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

Deriving Hamilton's equations from Lagrange's equations
Expressing Hamiltonian H(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 take-home assignment 7 (Simulation)

Examples of Hamiltonian mechanics in phase plots

1D Pendulum and phase plot (Simulation)

1D-HO phase-space control (Simulation)

Lecture 13. Poincare, Lagrange, Hamiltonian, and Jacobi mechanics (Unit 1 Ch. 12, Unit 2, Unit 3) (10.4.12)

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

Lecture 14. Complex Variables, Series, and Field Coordinates I. (Ch. 10 of Unit 1) (10.9.12)

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

Lecture 15. Complex Variables, Series, and Field Coordinates II. (Ch. 10 of Unit 1) (10.11.12)

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

Non-analytic 2D source field analysis

Lecture 16. Introducing GCC Lagrangian`a la Trebuchet Dynamics(Ch. 1-3 of Unit 2 and Unit 3)(10.18.12)

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 17. GCC Lagrange Equations for Trebuchet or "How do we ignore constraints?" (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 18. Hamilton Equations for Trebuchet and Other Things (Ch. 5-9 of Unit 2) 10.25.12

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 19. Reimann-Christoffel equations and covariant derivative(Ch. 4-7 of Unit 3) 10.30.12

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

Christoffel g-derivative formula

What's a tensor? What's not?

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

Small radial oscillations

Cycloid vs Pendulum

Lecture 20. Electromagnetic Lagrangian and charge-field mechanics (Ch. 2.8 of Unit 2) 11.1.12

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

Cycloid geometry and flying sticks

Lecture 21. Classical Constraints: Comparing various methods(Ch. 9 of Unit 3) 11.06.12

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 22. . Introduction to classical oscillation and resonance (Ch. 1 of Unit 4 11.08.12)

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/Γand 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)

Lecture 23. Introduction to coupled oscillation and eigenmodes (Ch. 2-4 of Unit 4 11.13.12)

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

Secular eq., Hamilton-Cayley eq., Idempotent projectors, (how eigenvalueseigenvectors)

Spectral decomposition and P-operator expansions (how projectorseigensolutions)

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

Lecture 24. Introduction to Spinor-Vector resonance dynamics (Ch. 2-4 of Unit 4 11.13.12)

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

Analogy: 2-State Schrodinger: iℏ∂t(t)=H|Ψ(t) versus Classical 2D-HO: t2x=-K•x

Hamilton-Pauli spinor symmetry ( σ-expansion in ABCD-Types) Hµσµ

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 25. Parametric Resonance and Multi-particle Wave Modes (Ch. 7-8 of Unit 4 11.27.12)

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 26. Introduction to Orbital Dynamics (Ch. 2-4 of Unit 5 11.29.12)

Orbits in Isotropic Harmonic Oscillator and Coulomb Potentials

Effective potentials for IHO and Coulomb orbits

Stable equilibrium radii and radial/angular frequency ratios

Classical turning radii and apogee/perigee parameters

Polar coordinate differential equations

Quadrature integration techniques

Detailed orbital functions

Relating orbital energy-momentum to conic-sectional orbital geometry

Kepler equation of time and phase geometry

Geometry and Symmetry of Coulomb orbits

Lecture 27. Geometry and Symmetry of Orbital Dynamics (Ch. 2-4 of Unit 5 12.04.12)

Geometry and Symmetry of Coulomb orbits

Detailed elliptic geometry

Detailed hyperbolic geometry

Rutherford scattering and differential scattering crossections

Ruler & compass construction

Eccentricity vector ε and orbital phase geometry (Lecture 27.5)

Ruler & compass construction

Lecture 28 Multi-particle and Rotational Dynamics (Ch. 2-7 of Unit 6 12.06.12)

2-Particle orbits

Copernican view

Ptolemetric view

2-Particle scattering Lab-vs.-Body frame views

Ruler & compass construction

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

Symmetric-top dynamics (Constant L)

BOD-frame cone rolling on LAB frame cone