Lecture 28 Mon.Lecture 12.02.2019 28 Multi-particle orbits and rotating body dynamics (Ch. 2-7 of Unit 6 12.07.17)

2-Particle orbits Separatrix circle pair RES contours dihedral angle KC~10 Ptolemetric or LAB view and reduced mass A-B θ =atan( ) 9 sep B-C Copernican or COM view and reduced coupling 8

7 π - θsep 2-Particle orbits and scattering: LAB-vs.-COM frame views 6 6 7 8 9 K ~10 θsep Ruler & compass construction (or not) 8 9 A

Rotational equivalent of Newton’s F=dp/dt equations: N=dL/dt -6 -7 How to make my boomerang come back -8 The gyrocompass and mechanical spin analogy -9 -10 Asym. Rotor AJP Rotational momentum and velocity tensor relations 44,11 1976 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 and Practical poolhall application This Lecture’s Reference Link Listing Web Resources - front page Quantum Theory for the Computer Age 2017 Group Theory for QM UAF Physics UTube channel Principles of Symmetry, Dynamics, and Spectroscopy 2018 Adv CM Classical Mechanics with a Bang! 2018 AMOP Modern Physics and its Classical Foundations 2019 Advanced Mechanics Lecture #22-28 In reverse order CMwBang Text 2012 Unit 6 page=5 Some Geometric Aspects of Classical Coulomb Scattering AJP 40 4 p1852 1972 BounceIt Web App/Scenarios: 5002, 5003 How Molecules do Self-NMR - Harter-Mitchell-Columbus-2009 CoulIt Web App/Scenarios: Classical Mechanics with a Bang! - Asymmetric Top Demo TwoParticleCollision_LToR, TwoParticleCollision_LToR_CM, TwoParticleOrbit_Coulomb, Allbookstores.com - Compare for Heller's SemiClassical Way - 0691163731 TwoParticleOrbit_Coulomb_CM, TwoParticleOrbit_Hooke, TwoParticleOrbit_Hooke_CM "My Bomerang Won't Come Back" (YouTube: Playlist) Singular Motion of Asymetric Rotators AJP 44, 11 p1080 Harter-Kim-1976 Rotating Solid Bodies in Microgravity (YouTube) Molecular Eigensolution Symmetry Analysis and Fine Structure - Harter-IJMS-2013 Dancing T-handle in zero-g (YouTube) Lenz Vector and Orbital Analog Computers - AJP 44 p348 1976 CoulIt Web App Simulations: p19, p32, p72, p73, p92, R=-0.375, R=+0.5, Rutherford Classical Mechanics with a Bang! 2018 OscillatorPE Web App: IHO Scenario 2, Coulomb Scenario 3 Lectures 8, 9, 23 page 93 RelaWavity Web App/Simulator/Calculator: Elliptical - IHO orbits Text Unit 6, page=27 JerkIt Web App: 2-, 2+, Amp50Omega147-, Amp50Omega296, Amp50Omega602, Gap(1) ColorU2 for the Web - in development MolVibes Web App: C3vN3 Group Theory for Quantum Mechanics - 2017 Lectures: 6, 7, 8, WaveIt Web App: and the combined 9-10 Dim = 3 w/Wave Components; Quantum Theory for the Computer Age Unit 3 Ch.7-10, page=90 Static Char Table: 6, 12, 12(b), 16, 36, 256 Spectral Decomposition with Repeated Eigenvalues - 2017 GTQM - Lecture 5 Quantum Carpet with N=20: Gaussian, Boxcar Web based 3D & XR (x∈{A,M,V}, R=Reality) https://www.babylonjs.com/ Quantum Revivals of Morse Oscillators and Farey-Ford Geometry - Li-Harter-CPL-2015 Web based 3D graphics WebGL API (Graphics Layer modeled after OpenGL) QTCA Unit_5 Ch14 2013 Lester. R. Ford, Am. Math. Monthly 45,586(1938) Recent In-House draft Articles: John Farey, Phil. Mag.(1816) Wolfram Springer handbook on Molecular Symmetry and Dynamics - Ch_32 - Molecular Symmetry AMOP Ch 0 Space-Time Symmetry - 2019 Harter, J. Mol. Spec. 210, 166-182 (2001) Seminar at Rochester Institute of Optics, Auxiliary slides, June 19, 2018 Harter, Li IMSS (2013) Li, Harter, Chem.Phys.Letters (2015) Quantum_Computing - (Current) State of the Art - Reimer-www-2019 Advanced Atomic and Molecular Optical Physics 2018 Class #9, pages: 5, 61 Geometric Algebra- A Guided Tour through Space and Time - Reimer-www-2019 BoxIt Web Simulations Wildlife Monitoring Identification and Behavioral Study - Section 1 - Reimer-www-2019 Wildlife Monitoring Identification and Behavioral Study - Section 2 - Reimer-www-2019 Pure A-Type A=4.9, B=0 ,C=0, & D=4.0 Pure B-Type: A=4.0, B=-0.2, C=0, & D=4.0 Quantum Computing (QC) and Geometric Algebra (GA) references: Pure C-Type A,D=4.055, B=0, C=0.1 Quantum_Supremacy_Using_a_Programmable_Superconducting_Processor_-_Arute-n-2019 Mixed AB-Type w/Cosine Quantum Computing for Computer Scientists - Helwer-mr-yt-2018, Slides Mixed AB Type A=4.0, BU2=0.866…, CU2=0, & D=1.0 w/Stokes & Freq rats Quantum Computing and Workforce, Curriculum, and App Devel - Roetteler-MS-2019 Mixed AB Type A=5.086 B=-0.27 C=0 D=2.024 w/Stokes plot Quantum_Computing - (Current) State of the Art - Reimer-www-2019 Mixed ABC Type A=4.833 B=0.2403 C=0.4162 D=4.277 w/Stokes plot Excerpts (Page 44-47 in Preliminary Draft) for a GA take on the Complex Numbers Recent mixed ABC Type A=0.325 B=0.375 C=0.825 D=0.05 w/Stokes plot Geometric Algebra- A Guided Tour through Space and Time - Reimer-www-2019 Select, exciting, and/or related Research GA & QC references (Page 11-16 in Preliminary Draft) This Indestructible NASA Camera Revealed Hidden Patterns on Jupiter - seeker-yt-2019 What did NASA's New Horizons discover around Pluto? - Astrum-yt-2018 Synthetic_Chiral_Light_for_Efficient_Control_of_Chiral_Light–Matter_Interaction_-_Ayuso-np-2019 In development, but close to role out. More Advanced QM and classical references will soon be available through our: References Page Continued for 3 more pages⤵ Would be great to have our Apache SOLR Search & Index system up for a bigger Bang!) This Lecture’s Reference Link Listing Web Resources - front page Quantum Theory for the Computer Age 2017 Group Theory for QM UAF Physics UTube channel Principles of Symmetry, Dynamics, and Spectroscopy 2018 Adv CM Classical Mechanics with a Bang! 2018 AMOP Modern Physics and its Classical Foundations 2019 Advanced Mechanics Lectures #12 through #21

Wiki on Pafnuty Chebyshev In reverse order Nobelprize.org Pirelli Relativity Challenge (Introduction level) - Visualizing Waves: 2005 Physics Award Using Earth as a clock, BoxIt Web Simulations: Tesla's AC Phasors , A-Type w/Cosine, A-Type w/Freq ratios, Phasors using complex numbers. AB-Type w/Cosine, AB-Type 2:1 Freq ratio CM wBang Unit 1 - Chapter 10, pdf_page=135 OscillIt Web Simulations: Calculus of exponentials, logarithms, and complex fields, Default/Generic, Weakly Damped #18, RelaWavity Web Simulation - Unit Circle and Hyperbola (Mixed labeling) Forced : Way below resonance,On resonance Smith Chart, Invented by Phillip H. Smith (1905-1987) Way above resonance,Underdamped Select, exciting, and related Research Complex Response Plot Clifford_Algebra_And_The_Projective_Model_Of_Homogeneous_Metric_Spaces - CoulIt Web Simulations: Foundations - Sokolov-x-2013 Stark-Coulomb : Bound-state motion in parabolic coordinates Geometric Algebra 3 - Complex Numbers - MacDonald-yt-2015 Molecular Ion : Bound-state motion in hyperbolic coordinates Biquaternion -Complexified Quaternion- Roots of -1 - Sangwine-x-2015 Synchrotron Motion, Synchrotron Motion #2 An_Introduction_to_Clifford_Algebras_and_Spinors_-_Vaz-Rocha-op-2016 Mechanical Analog to EM Motion (YouTube video) Unified View on Complex Numbers and Quaternions- Bongardt-wcmms-2015 iBall demo - Quasi-periodicity (YouTube video) Complex Functions and the Cauchy-Riemann Equations - complex2 - Friedman-columbia-2019 Trebuchet Web Simulations: An_sp-hybridized_Molecular_Carbon_Allotrope-_cyclo-18-carbon_-_Kaiser-s-2019 Default/Generic URL, Montezuma's Revenge, Seige of Kenilworth, An_Atomic-Scale_View_of_Cyclocarbon_Synthesis_-_Maier-s-2019 "Flinger", Discovery_Of_Topological_Weyl_Fermion_Lines_And_Drumhead_Surface_States_in_a_ Position Space (Course), Position Space (Fine) Room_Temperature_Magnet_-_Belopolski-s-2019 Wacky Waving Solid Metal Arm Flailing Chaos Pendulum - Scooba_Steeve-yt-2015 "Weyl"ing_away_Time-reversal_Symmetry_-_Neto-s-2019 Triple Double-Pendulum - Cohen-yt-2008 Non-Abelian_Band_Topology_in_Noninteracting_Metals_-_Wu-s-2019 Punkin Chunkin - TheArmchairCritic-2011 What_Industry_Can_Teach_Academia_-_Mao-s-2019 Jersey Team Claims Title in Punkin Chunkin - sussexcountyonline-1999 RoVib-_quantum_state_resolution_of_the_C60_fullerene_-_Changala-Ye-s-2019 (Alt) Shooting range for medieval siege weapons. Anybody knows? - twcenter.net/forums A_Degenerate_Fermi_Gas_of_Polar_molecules_-_DeMarco-s-2019 The Trebuchet - Chevedden-SciAm-1995 NOVA Builds a Trebuchet An assist from Physics Girl (YouTube Channel): How to Make VORTEX RINGS in a Pool Recent Articles of Interest: Crazy pool vortex - pg-yt-2014 A_Semi-Classical_Approach_to_the_Calculation_of_Highly_Excited_Rotational_Energies for … Fun with Vortex Rings in the Pool - pg-yt-2014 Asymmetric-Top_Molecules_-_Schmiedt-pccp-2017 Tunable and broadband coherent perfect absorption by ultrathin blk phos metasurfaces - Guo-josab-2019 Vortex Detection in Vector Fields Using Geometric Algebra - Pollock-aaca-2013.pdf Running Reference Link Listing Lectures #11 through #7

Eric J Heller Gallery: In reverse order Main portal, Consonance and Dissonance II, Bessel 21, Chladni CoulIt Web Simulations: Basic/Generic The Semiclassical Way to Molecular Spectroscopy - Heller-acs-1981 Exploding Starlet Quantum_dynamical_tunneling_in_bound_states_-_Davis-Heller- Volcanoes of Io (Color Quantized) jcp-1981 JerkIt Web Simulations: Basic/Generic Pendulum Web Simulation Catcher in the Eye - IHO with Linear Hooke perturbation - Force-potential-Velocity Plot Cycloidulum Web Simulation Links to previous lecture: Page=74, Page=75, Page=79 OscillatorPE Web Simulation: Pendulum Web Sim Coulomb-Newton-Inverse_Square, Cycloidulum Web Sim Hooke-Isotropic Harmonic, Pendulum-Circular_Constraint JerkIt Web Simulations: Basic/Generic: Inverted, FVPlot CMwithBang Lecture 8, page=20 AMOP Ch 0 Space-Time Symmetry - 2019 WWW.sciencenewsforstudents.org: Cassini - Saturnian polar vortex Seminar at Rochester Institute of Optics, Aux. slides-2018 “RelaWavity” Web Simulations: NASA Astronomy Picture of the Day - 2-CW laser wave, Lagrangian vs Hamiltonian, Io: The Prometheus Plume (Just Image) Physical Terms Lagrangian L(u) vs Hamiltonian H(p) NASA Galileo - Io's Alien Volcanoes CoulIt Web Simulation of the Volcanoes of Io New Horizons - Volcanic Eruption Plume on Jupiter's moon IO BohrIt Multi-Panel Plot: NASA Galileo - A Hawaiian-Style Volcano on Io Relativistically shifted Time-Space plots of 2 CW light waves Pirelli Site: Phasors animimation BoxIt Web Simulations: CMwithBang Lecture #6, page=70 (9.10.18) Generic/Default Most Basic A-Type Select, exciting, and related Research & Articles of Interest: Basic A-Type w/reference lines Burning a hole in reality—design for a new laser may be powerful enough to pierce space-time - Sumner-KOS-2019 Basic A-Type A-Type with Potential energy Trampoline mirror may push laser pulse through fabric of the Universe - Lee-ArsTechnica-2019 A-Type with Potential energy and Stokes Plot Achieving_Extreme_Light_Intensities_using_Optically_Curved_Relativistic_Plasma_Mirrors_-_Vincenti-prl-2019 A-Type w/3 time rates of change A_Soft_Matter_Computer_for_Soft_Robots_-_Garrad-sr-2019 A-Type w/3 time rates of change with Stokes Plot Correlated_Insulator_Behaviour_at_Half-Filling_in_Magic-Angle_Graphene_Superlattices_-_cao-n-2018 B-Type (A=1.0, B=-0.05, C=0.0, D=1.0) Sorting_ultracold_atoms_in_a_three-dimensional_optical_lattice_in_a_ realization_of_Maxwell's_Demon - Kumar-n-2018 RelaWavity Web Elliptical Motion Simulations: Synthetic_three-dimensional_atomic_structures_assembled_atom_by_atom - Barredo-n-2018 Orbits with b/a=0.125 Older ones: Orbits with b/a=0.5 Wave-particle_duality_of_C60_molecules - Arndt-ltn-1999 Orbits with b/a=0.7 Optical_Vortex_Knots_-_One_Photon__At_A_Time - Tempone-Wiltshire-Sr-2018 Exegesis with b/a=0.125 Baryon_Deceleration_by_Strong_Chromofields_in_Ultrarelativistic_, Exegesis with b/a=0.5 Nuclear_Collisions - Mishustin-PhysRevC-2007, APS Link & Abstract Exegesis with b/a=0.7 Hadronic Molecules - Guo-x-2017 Contact Ellipsometry Hidden-charm_pentaquark_and_tetraquark_states - Chen-pr-2016 Running Reference Link Listing Lectures #6 through #1 In reverse order BounceIt Web Animation - Scenarios: RelaWavity Web Simulation: Contact Ellipsometry Generic Scenario: 2-Balls dropped no Gravity (7:1) - V vs V Plot (Power=4) BoxIt Web Simulation: Elliptical Motion (A-Type) 1-Ball dropped w/Gravity=0.5 w/Potential Plot: Power=1, Power=4 CMwBang Course: Site Title Page 7:1 - V vs V Plot: Power=1 Pirelli Relativity Challenge: Describing Wave Motion With Complex Phasors 3-Ball Stack (10:3:1) w/Newton plot (y vs t) - Power=4 UAF Physics UTube channel 3-Ball Stack (10:3:1) w/Newton plot (y vs t) - Power=1 3-Ball Stack (10:3:1) w/Newton plot (y vs t) - Power=1 w/Gaps Velocity Amplification in Collision Experiments Involving Superballs - Harter, 1971 4-Ball Stack (27:9:3:1) w/Newton plot (y vs t) - Power=4 MIT OpenCourseWare: High School/Physics/Impulse and Momentum 4-Newton's Balls (1:1:1:1) w/Newtonian plot (y vs t) - Power=4 w/Gaps Hubble Site: Supernova - SN 1987A 6-Ball Totally Inelastic (1:1:1:1:1:1) w/Gaps: Newtonian plot (t vs x), V6 vs V5 plot 5-Ball Totally Inelastic Pile-up w/ 5-Stationary-Balls - Minkowski plot (t vs x1) w/Gaps 1-Ball Totally Inelastic Pile-up w/ 5-Stationary-Balls - Vx2 vs Vx1 plot w/Gaps BounceItIt Web Animation - Scenarios: BounceIt Dual plots 49:1 y vs t, 49:1 V2 vs V1, 1:500:1 - 1D Gas Model w/ faux restorative force (Cool), m1:m2 = 3:1 1:500:1 - 1D Gas (Warm), 1:500:1 - 1D Gas Model (Cool, Zoomed in), v2 vs v1 and V2 vs V1, (v1, v2)=(1, 0.1), (v1, v2)=(1, 0) Farey Sequence - Wolfram y2 vs y1 plots: (v1, v2)=(1, 0.1), (v1, v2)=(1, 0), (v1, v2)=(1, -1) Fractions - Ford-AMM-1938 Estrangian plot V2 vs V1: (v1, v2)=(0, 1), (v1, v2)=(1, -1) Monstermash BounceItIt Animations: m1:m2 = 4:1 1000:1 - V2 vs V1, 1000:1 with t vs x - Minkowski Plot v2 vs v1, y2 vs y1 Quantum Revivals of Morse Oscillators and Farey-Ford Geometry - Li-Harter-2013 m1:m2 = 100:1, (v1, v2)=(1, 0): V2 vs V1 Estrangian plot, y2 vs y1 plot Quantum_Revivals_of_Morse_Oscillators_and_Farey-Ford_Geometry - Li-Harter-cpl-2015 Quant. Revivals of Morse Oscillators and Farey-Ford Geom. - Harter-Li-CPL-2015 (Publ.) With g=0 and 70:10 mass ratio Velocity_Amplification_in_Collision_Experiments_Involving_Superballs-Harter-1971 With non zero g, velocity dependent damping and mass ratio of 70:35 WaveIt Web Animation - Scenarios: M1=49, M2=1 with Newtonian time plot Quantum_Carpet, Quantum_Carpet_wMBars, M1=49, M2=1 with V2 vs V1 plot Quantum_Carpet_BCar, Quantum_Carpet_BCar_wMBars Example with friction Wave Node Dynamics and Revival Symmetry in Quantum Rotors - Harter-JMS-2001 Low force constant with drag displaying a Pass-thru, Fall-Thru, Bounce-Off Wave Node Dynamics and Revival Symmetry in Quantum Rotors - Harter-jms-2001 (Publ.) m1:m2= 3:1 and (v1, v2) = (1, 0) Comparison with Estrangian

X2 paper: Velocity Amplification in Collision Experiments Involving Superballs - Harter, et. al. 1971 (pdf) Car Collision Web Simulator: https://modphys.hosted.uark.edu/markup/CMMotionWeb.html AJP article on superball dynamics Superball Collision Web Simulator: https://modphys.hosted.uark.edu/markup/BounceItWeb.html; with Scenarios: 1007 AAPT Summer Reading List BounceIt web simulation with g=0 and 70:10 mass ratio Scitation.org - AIP publications With non zero g, velocity dependent damping and mass ratio of 70:35 HarterSoft Youtube Channel Elastic Collision Dual Panel Space vs Space: Space vs Time (Newton) , Time vs. Space(Minkowski) Inelastic Collision Dual Panel Space vs Space: Space vs Time (Newton), Time vs. Space(Minkowski) Matrix Collision Simulator:M1=49, M2=1 V2 vs V1 plot <> 2-Particle orbits and center-of-mass (CM) coordinate frame

m2 r = r1- r2 r = m r + m r r2 rCM m CM 1 1 2 2 r 1 1 m1+m2

Defining relative coordinate vector

r = r − r 1 2 and mass-weighted-average or center-of-mass coordinate vector rCM m r + m r r = r = 1 1 2 2 CM m + m 1 2 The inverse coordinate transformation. m r m r r = r + 2 , r = r − 1 1 CM m + m 2 CM m + m 1 2 1 2 2-Particle orbits Ptolemetric or LAB view and reduced mass Copernican or COM view and reduced coupling Reduced mass: Ptolemetric views

Radial inter-particle force F12 is on m1 due to m2 and F21 = -F12 is on m2 due to m1 r F(r) F12 = F(r)er =-F21 = F(r)rˆ = F(r) = r − r r r ( 1 2 ) r F(r) F = m !r! = F(r)rˆ = F(r) = r − r F12 acts along relative coordinate vector r= r1 - r2 12 1 1 r r ( 1 2 ) Depends only upon the relative distance r =| r1 - r2 | r F(r) F = m !r! = −F(r)rˆ = −F(r) = − r − r 21 2 2 r r ( 1 2 )

m r µ −m r −µ Re-scaled force: A Copernican view r = 2 = r , r = 1 = r 1 m + m m 2 m + m m relative radius vector 1 2 1 1 2 2 m1 −m2 r1 = r = r2 µ µ m2 r = r1- r2 r = m r + m r r2 rCM m CM 1 1 2 2 m r m r r 1 2 1 1 m1+m2 r1 = rCM + , r2 = rCM − m1 + m2 m1 + m2 Reduced mass: Ptolemetric views

Radial inter-particle force F12 is on m1 due to m2 and F21 = -F12 is on m2 due to m1 r F(r) F12 = F(r)er =-F21 = F(r)rˆ = F(r) = r − r r r ( 1 2 ) r F(r) F = m !r! = F(r)rˆ = F(r) = r − r F12 acts along relative coordinate vector r= r1 - r2 12 1 1 r r ( 1 2 ) Depends only upon the relative distance r =| r1 - r2 | r F(r) F = m !r! = −F(r)rˆ = −F(r) = − r − r 21 2 2 r r ( 1 2 ) Sum F12+F21 yields zero because of Newton's 3rd -law action-reaction cancellation. (m + m )!r! = m !r! + m !r! = 0 (a.k.a: Momentm Conservation) 1 2 CM 1 1 2 2

m r µ −m r −µ Re-scaled force: A Copernican view r = 2 = r , r = 1 = r 1 m + m m 2 m + m m relative radius vector 1 2 1 1 2 2 m1 −m2 r1 = r = r2 µ µ m2 r = r1- r2 r = m r + m r r2 rCM m CM 1 1 2 2 m r m r r 1 2 1 1 m1+m2 r1 = rCM + , r2 = rCM − m1 + m2 m1 + m2 Reduced mass: Ptolemetric views

Radial inter-particle force F12 is on m1 due to m2 and F21 = -F12 is on m2 due to m1 r F(r) F12 = F(r)er =-F21 = F(r)rˆ = F(r) = r − r r r ( 1 2 ) r F(r) F = m !r! = F(r)rˆ = F(r) = r − r F12 acts along relative coordinate vector r= r1 - r2 12 1 1 r r ( 1 2 ) Depends only upon the relative distance r =| r1 - r2 | r F(r) F = m !r! = −F(r)rˆ = −F(r) = − r − r 21 2 2 r r ( 1 2 ) Sum F12+F21 yields zero because of Newton's 3rd -law action-reaction cancellation. (m + m )!r! = m !r! + m !r! = 0 (a.k.a: Momentm Conservation) 1 2 CM 1 1 2 2

m m r F(r) µ = 2 1 Difference F12-F21 reduces to µ ! ! = using reduced mass: !r!CM = 0 m1 + m2

m r µ −m r −µ Re-scaled force: A Copernican view r = 2 = r , r = 1 = r 1 m + m m 2 m + m m relative radius vector 1 2 1 1 2 2 m1 −m2 r1 = r = r2 µ µ m2 r = r1- r2 r = m r + m r r2 rCM m CM 1 1 2 2 m r m r r 1 2 1 1 m1+m2 r1 = rCM + , r2 = rCM − m1 + m2 m1 + m2 Reduced mass: Ptolemetric views

Radial inter-particle force F12 is on m1 due to m2 and F21 = -F12 is on m2 due to m1 r F(r) F12 = F(r)er =-F21 = F(r)rˆ = F(r) = r − r r r ( 1 2 ) r F(r) F = m !r! = F(r)rˆ = F(r) = r − r F12 acts along relative coordinate vector r= r1 - r2 12 1 1 r r ( 1 2 ) Depends only upon the relative distance r =| r1 - r2 | r F(r) F = m !r! = −F(r)rˆ = −F(r) = − r − r 21 2 2 r r ( 1 2 ) Sum F12+F21 yields zero because of Newton's 3rd -law action-reaction cancellation. (m + m )!r! = m !r! + m !r! = 0 1 2 CM 1 1 2 2

m2m1 Difference F12-F21 reduces to µ !r ! = F ( r ) using reduced mass: µ = and: !r! = 0 m + m CM 2F(r) 1 2 [ m1!r!1 ]−[ m2!r!2 ] = (r1 − r2 ) r

m r µ −m r −µ Re-scaled force: A Copernican view r = 2 = r , r = 1 = r 1 m + m m 2 m + m m relative radius vector 1 2 1 1 2 2 m1 −m2 r1 = r = r2 µ µ m2 r = r1- r2 r = m r + m r r2 rCM m CM 1 1 2 2 m r m r r 1 2 1 1 m1+m2 r1 = rCM + , r2 = rCM − m1 + m2 m1 + m2 Reduced mass: Ptolemetric views

Radial inter-particle force F12 is on m1 due to m2 and F21 = -F12 is on m2 due to m1 r F(r) F12 = F(r)er =-F21 = F(r)rˆ = F(r) = r − r r r ( 1 2 ) r F(r) F = m !r! = F(r)rˆ = F(r) = r − r F12 acts along relative coordinate vector r= r1 - r2 12 1 1 r r ( 1 2 ) Depends only upon the relative distance r =| r1 - r2 | r F(r) F = m !r! = −F(r)rˆ = −F(r) = − r − r 21 2 2 r r ( 1 2 ) Sum F12+F21 yields zero because of Newton's 3rd -law action-reaction cancellation. (m + m )!r! = m !r! + m !r! = 0 1 2 CM 1 1 2 2

m2m1 Difference F12-F21 reduces to µ !r ! = F ( r ) using reduced mass: µ = and: !r! = 0 m + m CM 2F(r) 1 2 [ m1!r!1 ]−[ m2!r!2 ] = (r1 − r2 ) r

⎡ m1m2!r! ⎤ ⎡ m2m1!r! ⎤ 2F(r) ⎢m1!r!CM + ⎥ − ⎢m2!r!CM − ⎥= (r1 − r2 ) ⎣ m1 + m2 ⎦ ⎣ m1 + m2 ⎦ r

gives: µ !r! = F(r)rˆ = F(r)e = F(r) r m r µ −m r −µ Re-scaled force: A Copernican view r = 2 = r , r = 1 = r 1 m + m m 2 m + m m relative radius vector 1 2 1 1 2 2 m1 −m2 r1 = r = r2 µ µ m2 r = r1- r2 r = m r + m r r2 rCM m CM 1 1 2 2 m r m r r 1 2 1 1 m1+m2 r1 = rCM + , r2 = rCM − m1 + m2 m1 + m2 Reduced mass: Ptolemetric views

Radial inter-particle force F12 is on m1 due to m2 and F21 = -F12 is on m2 due to m1 r F(r) F12 = F(r)er =-F21 = F(r)rˆ = F(r) = r − r r r ( 1 2 ) r F(r) F = m !r! = F(r)rˆ = F(r) = r − r F12 acts along relative coordinate vector r= r1 - r2 12 1 1 r r ( 1 2 ) Depends only upon the relative distance r =| r1 - r2 | r F(r) F = m !r! = −F(r)rˆ = −F(r) = − r − r 21 2 2 r r ( 1 2 ) Sum F12+F21 yields zero because of Newton's 3rd -law action-reaction cancellation. (m + m )!r! = m !r! + m !r! = 0 1 2 CM 1 1 2 2

m2m1 Difference F12-F21 reduces to µ !r ! = F ( r ) using reduced mass: µ = and: !r! = 0 m + m CM 2F(r) 1 2 [ m1!r!1 ]−[ m2!r!2 ] = (r1 − r2 ) r

⎡ m1m2!r! ⎤ ⎡ m2m1!r! ⎤ 2F(r) ⎢m1!r!CM + ⎥ − ⎢m2!r!CM − ⎥= (r1 − r2 ) ⎣ m1 + m2 ⎦ ⎣ m1 + m2 ⎦ r

2m1m2!r! 2F(r) = (r1 − r2 ) gives: µ !r! = F(r)rˆ = F(r)e = F(r) m1 + m2 r r m r µ −m r −µ Re-scaled force: A Copernican view r = 2 = r , r = 1 = r 1 m + m m 2 m + m m relative radius vector 1 2 1 1 2 2 m1 −m2 r1 = r = r2 µ µ m2 r = r1- r2 r = m r + m r r2 rCM m CM 1 1 2 2 m r m r r 1 2 1 1 m1+m2 r1 = rCM + , r2 = rCM − m1 + m2 m1 + m2 Reduced mass: Ptolemetric views

Radial inter-particle force F12 is on m1 due to m2 and F21 = -F12 is on m2 due to m1 r F(r) F12 = F(r)er =-F21 = F(r)rˆ = F(r) = r − r r r ( 1 2 ) r F(r) F = m !r! = F(r)rˆ = F(r) = r − r F12 acts along relative coordinate vector r= r1 - r2 12 1 1 r r ( 1 2 ) Depends only upon the relative distance r =| r1 - r2 | r F(r) F = m !r! = −F(r)rˆ = −F(r) = − r − r 21 2 2 r r ( 1 2 ) Sum F12+F21 yields zero because of Newton's 3rd -law action-reaction cancellation. (m + m )!r! = m !r! + m !r! = 0 1 2 CM 1 1 2 2

m2m1 Difference F12-F21 reduces to µ !r ! = F ( r ) using reduced mass: µ = and: !r! = 0 m + m CM 2F(r) 1 2 ⎛ ⎞ [ m1!r!1 ]−[ m2!r!2 ] = (r1 − r2 ) m ⎜ m ⎟ r µ = 2 = m ⎜1 − 2 ...⎟ (m >>m ) 1 1 1 m + m 2 ⎜ ⎟ 1 2 1 2 m ⎝⎜ m ⎠⎟ ⎡ m1m2!r! ⎤ ⎡ m2m1!r! ⎤ 2F(r) = + = 1 + 2 1 ⎢m1!r!CM + ⎥ − ⎢m2!r!CM − ⎥= (r1 − r2 ) µ m m m m (Why it’s reduced) m + m m + m r 1 2 1 2 m1 ⎣ 1 2 ⎦ ⎣ 1 2 ⎦ ⎛ ⎞ m ⎜ m ⎟ 1 m ⎜ 1 ⎟ m m µ = = 1 ⎜1 − ...⎟ ( 2>> 1) 2m1m2!r! 2F(r) m ⎜ m ⎟ = (r1 − r2 ) 1 ⎝ 2 ⎠ m + m r gives: µ !r! = F(r)rˆ = F(r)e = F(r) 1 + 1 2 r m 2 m r µ −m r −µ Re-scaled force: A Copernican view r = 2 = r , r = 1 = r 1 m + m m 2 m + m m relative radius vector 1 2 1 1 2 2 m1 −m2 r1 = r = r2 µ µ m2 r = r1- r2 r = m r + m r r2 rCM m CM 1 1 2 2 m r m r r 1 2 1 1 m1+m2 r1 = rCM + , r2 = rCM − m1 + m2 m1 + m2 2-Particle orbits Ptolemetric view and reduced mass Copernican view and reduced coupling Reduced mass: Ptolemetric views

Radial inter-particle force F12 is on m1 due to m2 and F21 = -F12 is on m2 due to m1 r F(r) F12 = F(r)er =-F21 = F(r)rˆ = F(r) = r − r r r ( 1 2 ) r F(r) F = m !r! = F(r)rˆ = F(r) = r − r F12 acts along relative coordinate vector r= r1 - r2 12 1 1 r r ( 1 2 ) Depends only upon the relative distance r =| r1 - r2 | r F(r) F = m !r! = −F(r)rˆ = −F(r) = − r − r 21 2 2 r r ( 1 2 ) Sum F12+F21 yields zero because of Newton's 3rd -law action-reaction cancellation. (m + m )!r! = m !r! + m !r! = 0 1 2 CM 1 1 2 2

m2m1 Difference F12-F21 reduces to µ !r ! = F ( r ) using reduced mass: µ = and: !r! = 0 m + m CM 2F(r) 1 2 ⎛ ⎞ [ m1!r!1 ]−[ m2!r!2 ] = (r1 − r2 ) m ⎜ m ⎟ r µ = 2 = m ⎜1 − 2 ...⎟ (m >>m ) 1 1 1 m + m 2 ⎜ ⎟ 1 2 1 2 m ⎝⎜ m ⎠⎟ ⎡ m1m2!r! ⎤ ⎡ m2m1!r! ⎤ 2F(r) = + = 1 + 2 1 ⎢m1!r!CM + ⎥ − ⎢m2!r!CM − ⎥= (r1 − r2 ) µ m m m m (Why it’s reduced) m + m m + m r 1 2 1 2 m1 ⎣ 1 2 ⎦ ⎣ 1 2 ⎦ ⎛ ⎞ m ⎜ m ⎟ 1 m ⎜ 1 ⎟ m m µ = = 1 ⎜1 − ...⎟ ( 2>> 1) 2m1m2!r! 2F(r) m ⎜ m ⎟ = (r1 − r2 ) 1 ⎝ 2 ⎠ m + m r gives: µ !r! = F(r)rˆ = F(r)e = F(r) 1 + 1 2 r m 2 m r µ −m r −µ Re-scaled force: A Copernican view r = 2 = r , r = 1 = r 1 m + m m 2 m + m m relative radius vector 1 2 1 1 2 2 m1 −m2 r1 = r = r2 µ µ (Here we get “reduced” coupling constants) each particle keeps it original mass m1 or m2, but feels coordinate-re-scaled force field F(m1 r1/µ) or F(m2 r2/µ) field m F = m !r! = F( 1 r )rˆ = −F 12 1 1 µ 1 1 21 m F = m !r! = F( 2 r )rˆ = −F 21 2 2 µ 2 2 12 Reduced mass: Ptolemetric views

Radial inter-particle force F12 is on m1 due to m2 and F21 = -F12 is on m2 due to m1 r F(r) F12 = F(r)er =-F21 = F(r)rˆ = F(r) = r − r r r ( 1 2 ) r F(r) F = m !r! = F(r)rˆ = F(r) = r − r F12 acts along relative coordinate vector r= r1 - r2 12 1 1 r r ( 1 2 ) Depends only upon the relative distance r =| r1 - r2 | r F(r) F = m !r! = −F(r)rˆ = −F(r) = − r − r 21 2 2 r r ( 1 2 ) Sum F12+F21 yields zero because of Newton's 3rd -law action-reaction cancellation. (m + m )!r! = m !r! + m !r! = 0 1 2 CM 1 1 2 2

m2m1 Difference F12-F21 reduces to µ !r ! = F ( r ) using reduced mass: µ = and: !r! = 0 m + m CM 2F(r) 1 2 ⎛ ⎞ [ m1!r!1 ]−[ m2!r!2 ] = (r1 − r2 ) 1 1 1 m + m m ⎜ m ⎟ r 1 2 µ = 2 = m ⎜1 − 2 ...⎟ (m >>m ) = + = 2 ⎜ ⎟ 1 2 m m m m m2 ⎝⎜ m1 ⎠ ⎡ m1m2!r! ⎤ ⎡ m2m1!r! ⎤ 2F(r) µ 1 2 1 2 1 + ⎢m1!r!CM + ⎥ − ⎢m2!r!CM − ⎥= (r1 − r2 ) (Why it’s reduced) m + m m + m r m1 ⎣ 1 2 ⎦ ⎣ 1 2 ⎦ ⎛ ⎞ m ⎜ m ⎟ 1 m ⎜ 1 ⎟ m m µ = = 1 ⎜1 − ...⎟ ( 2>> 1) 2m1m2!r! 2F(r) m ⎜ m ⎟ = (r1 − r2 ) 1 ⎝ 2 ⎠ m + m r gives: µ !r! = F(r)rˆ = F(r)e = F(r) 1 + 1 2 r m 2 m r µ −m r −µ Re-scaled force: A Copernican view r = 2 = r , r = 1 = r 1 m + m m 2 m + m m relative radius vector 1 2 1 1 2 2 m1 −m2 r1 = r = r2 µ µ (Here we get “reduced” coupling constants) each particle keeps it original mass m1 or m2, but feels coordinate-re-scaled force field F(m1 r1/µ) or F(m2 r2/µ) field m k m µ2 k F = m !r! = F( 1 r )rˆ = −F F(r) = becomes: F( 1 r ) = 12 1 1 µ 1 1 21 2 µ 1 2 2 r (Coulomb) m1 r1 m2 2 2 2 2 F21 = m2!r!2 = F( r2 )rˆ2 = −F12 k → k = k µ / m , k → k = k µ / m µ 1 1 2 2 Reduced mass: Ptolemetric views

Radial inter-particle force F12 is on m1 due to m2 and F21 = -F12 is on m2 due to m1 r F(r) F12 = F(r)er =-F21 = F(r)rˆ = F(r) = r − r r r ( 1 2 ) r F(r) F = m !r! = F(r)rˆ = F(r) = r − r F12 acts along relative coordinate vector r= r1 - r2 12 1 1 r r ( 1 2 ) Depends only upon the relative distance r =| r1 - r2 | r F(r) F = m !r! = −F(r)rˆ = −F(r) = − r − r 21 2 2 r r ( 1 2 ) Sum F12+F21 yields zero because of Newton's 3rd -law action-reaction cancellation. (m + m )!r! = m !r! + m !r! = 0 1 2 CM 1 1 2 2

m2m1 Difference F12-F21 reduces to µ !r ! = F ( r ) using reduced mass: µ = and: !r! = 0 m + m CM 2F(r) 1 2 ⎛ ⎞ [ m1!r!1 ]−[ m2!r!2 ] = (r1 − r2 ) 1 1 1 m + m m ⎜ m ⎟ r 1 2 µ = 2 = m ⎜1 − 2 ...⎟ (m >>m ) = + = 2 ⎜ ⎟ 1 2 m m m m m2 ⎝⎜ m1 ⎠ ⎡ m1m2!r! ⎤ ⎡ m2m1!r! ⎤ 2F(r) µ 1 2 1 2 1 + ⎢m1!r!CM + ⎥ − ⎢m2!r!CM − ⎥= (r1 − r2 ) (Why it’s reduced) m + m m + m r m1 ⎣ 1 2 ⎦ ⎣ 1 2 ⎦ ⎛ ⎞ m ⎜ m ⎟ 1 m ⎜ 1 ⎟ m m µ = = 1 ⎜1 − ...⎟ ( 2>> 1) 2m1m2!r! 2F(r) m ⎜ m ⎟ = (r1 − r2 ) 1 ⎝ 2 ⎠ m + m r gives: µ !r! = F(r)rˆ = F(r)e = F(r) 1 + 1 2 r m 2 m r µ −m r −µ Re-scaled force: A Copernican view r = 2 = r , r = 1 = r 1 m + m m 2 m + m m relative radius vector 1 2 1 1 2 2 m1 −m2 r1 = r = r2 µ µ (Here we get “reduced” coupling constants) each particle keeps it original mass m1 or m2, but feels coordinate-re-scaled force field F(m1 r1/µ) or F(m2 r2/µ) field m k m µ2 k m m F m r F( 1 r )rˆ F 1 1 1 12 = 1!!1 = 1 1 = − 21 F(r) = becomes: F( r1) = F(r) = −kr becomes: F( r ) = − k r µ r2 µ m 2 r 2 1 1 (Coulomb) 1 1 (Harmonic Oscillator) µ µ m2 2 2 2 2 F21 = m2!r!2 = F( r2 )rˆ2 = −F12 k → k = k µ / m , k → k = k µ / m k → k =k m /µ , k → k =k m /µ µ 1 1 2 2 1 1 2 2 2-Particle orbits and scattering: LAB-vs.-COM frame views Ruler & compass construction (or not) Examples of Coulomb and harmonic oscillator 2-particle “Copernican” orbits in CM system.

(a) F(r) = -k/r2 (b) F(r) = -kr m2

r2 rCM= 0 m2

r2

CoulIt Web Simulations r1 CoulIt Web Simulations Coulombic Orbit (CM Frame) Hooke Orbit (CM Frame) r1 Coulombic Orbit (Lab Frame) Hooke Orbit (Lab Frame) m1 m1 Two particles are in synchronous motion around fixed CM origin. Orbit periods are identical to each other. Orbits are mass-scaled copies with equal aspect ratio (a/b), eccentricity, and orientation.

Examples of Coulomb and harmonic oscillator 2-particle “Copernican” orbits in CM system.

(a) F(r) = -k/r2 (b) F(r) = -kr m2

r2 rCM= 0 m2

r2

CoulIt Web Simulations r1 CoulIt Web Simulations Coulombic Orbit (CM Frame) Hooke Orbit (CM Frame) r1 Coulombic Orbit (Lab Frame) Hooke Orbit (Lab Frame) m1 m1 Two particles are in synchronous motion around fixed CM origin. Orbit periods are identical to each other. Orbits are mass-scaled copies with equal aspect ratio (a/b), eccentricity, and orientation. Orbits differ in size of axes (a1 , b1) and (a2 , b2) Orbits differ in placement of center (for the Coulomb case) or foci (for the oscillator).

Examples of Coulomb and harmonic oscillator 2-particle “Copernican” orbits in CM system.

(a) F(r) = -k/r2 (b) F(r) = -kr m2

r2 rCM= 0 m2

r2

CoulIt Web Simulations r1 CoulIt Web Simulations Coulombic Orbit (CM Frame) Hooke Orbit (CM Frame) r1 Coulombic Orbit (Lab Frame) Hooke Orbit (Lab Frame) m1 m1 Two particles are in synchronous motion around fixed CM origin. Orbit periods are identical to each other. Orbits are mass-scaled copies with equal aspect ratio (a/b), eccentricity, and orientation. Orbits differ in size of axes (a1 , b1) and (a2 , b2) Orbits differ in placement of center (for the Coulomb case) or foci (for the oscillator). Orbit axial dimensions (ak , bk) and λk are in inverse proportion to mass values. a m = a m = a µ , b m = b m = bµ λ m = λ m = λµ 1 1 2 2 1 1 2 2 1 1 2 2 Examples of Coulomb and harmonic oscillator 2-particle “Copernican” orbits in CM system.

(a) F(r) = -k/r2 (b) F(r) = -kr m2

r2 rCM= 0 m2

r2

CoulIt Web Simulations r1 CoulIt Web Simulations Coulombic Orbit (CM Frame) Hooke Orbit (CM Frame) r1 Coulombic Orbit (Lab Frame) Hooke Orbit (Lab Frame) m1 m1 Two particles are in synchronous motion around fixed CM origin. Orbit periods are identical to each other. Orbits are mass-scaled copies with equal aspect ratio (a/b), eccentricity, and orientation. Orbits differ in size of axes (a1 , b1) and (a2 , b2) Orbits differ in placement of center (for the Coulomb case) or foci (for the oscillator). Orbit axial dimensions (ak , bk) and λk are in inverse proportion to mass values. a m = a m = a µ , b m = b m = bµ λ m = λ m = λµ 1 1 2 2 1 1 2 2 1 1 2 2 Harmonic oscillator periods and Coulomb orbit periods and eccentricity must match

m m 3 3 3 µ 1 2 µ a m1a1 m2a2 TIHO = 2π = 2π = 2π T = 2π = 2π = 2π ε = ε = ε k k k Coul k k k 1 2 1 2 1 2 Examples of Coulomb and harmonic oscillator 2-particle “Copernican” orbits in CM system.

(a) F(r) = -k/r2 (b) F(r) = -kr m2

r2 rCM= 0 m2

r2

CoulIt Web Simulations Coulombic r1 CoulIt Web Simulations Hooke Orbit Orbit (CM Frame) (CM Frame) r1 Coulombic Orbit (Lab Frame) Hooke Orbit (Lab Frame) m1 m1 Two particles are in synchronous motion around fixed CM origin. Orbit periods are identical to each other. Orbits are mass-scaled copies with equal aspect ratio (a/b), eccentricity, and orientation. Orbits differ in size of axes (a1 , b1) and (a2 , b2) Orbits differ in placement of center (for the Coulomb case) or foci (for the oscillator). Orbit axial dimensions (ak , bk) and λk are in inverse proportion to mass values. a m = a m = a µ , b m = b m = bµ λ m = λ m = λµ 1 1 2 2 1 1 2 2 1 1 2 2 Harmonic oscillator periods and Coulomb orbit periods and eccentricity must match

m m 3 3 3 µ 1 2 µ a m1a1 m2a2 TIHO = 2π = 2π = 2π T = 2π = 2π = 2π ε = ε = ε k k k Coul k k k 1 2 1 2 1 2 Three Coulomb orbit energy values satisfy the same proportion relation as their axes k k k E m = E m = Eµ , where: E = 1 , E = 2 , E = . 1 1 2 2 1 2a 2 2a 2a 1 2 Energy values and axes satisfy similar sum relations m1 m2 m1 m2 E1 + E2 = E + E = E , and: a1 + a2 = a + a = a µ µ µ µ A common type of scattering (m1=m2) ...that every pool shark should know Θ1LAB

90°

8

Θ2LAB

8 CM view LAB view CM vCM= -v CM(0) v =0 2 LAB LAB Θ1 v2 (0)=0

QCM CM CM Θ v CM(∞) LAB v1 (∞) 1 v1 (∞)

v CM(0) v CM(0) CM CM 2 1 v2 (0) 90° v1 (0)

CM v (∞) LAB v CM(∞) LAB 2 v2 (∞) 2 Θ2

BounceIt Web Simulations Hard LAB Collision (CM Frame) v2 (0)=0 Hard Collision (Lab Frame) To transform CM to LAB frame (Said: Galileo) Just subtract v2CM(0) from all v ΘCM (Assuming that initial v2LAB(0) is zero so v2CM(0) is CM velocity in LAB) CM view vCM=0

v CM(∞) 1 ΘCM m1 a1 r 1 rCM= 0 CM b1 v2 (0) CM v1 (0) b r 2 CM 2 v2 (∞) a2 m2 ΘCM ΘCM

CoulIt Web Simulation - Coulombic Collision (CM Frame)

LAB view vCM= -v CM(0) m1 2 LAB LAB Θ1 v2 (0)=0 Θ1LAB v CM(∞) 1 ΘCM v LAB(∞) m1 COG moves uniformly 1 CM CM at v = -v (0) 2 CM v2 (0) CM v1 (0) LAB m m 2 2 v LAB(∞) v2 (0)=0 2 CM Θ LAB v2 (∞) (m "creeps" logarithmically 2 LAB LAB 2 v2 (0)=0 Θ2 from -∞ until it is hit by m1)

m2

CoulIt Web Simulation - Coulombic Collision (LAB Frame) From: Geometric aspects of classical Coulomb scattering American Journal of Physics 40,1852-1856 (1972) Class project when I taught Jr. CM at Georgia Tech (Just 5 students) 27 The trouble with the Coulomb field is... ∫ t −1dt = lnt + C

F vLAB (t) = dt Coulomb 2 ∫ m2 k ≅ dt Coulomb ∫ CM 2 m2 ⎣⎡v1 (initial)⋅t ⎦⎤ − k 1 ≅ dt Coulomb ∼ − ln(t) ∫ CM 2 t m2 ⎣⎡v1 (initial)⎦⎤

From: Geometric aspects of classical Coulomb scattering American Journal of Physics 40,1852-1856 (1972) Class project when I taught Jr. CM at Georgia Tech (Just 5 students) From: Geometric aspects of classical Coulomb scattering American Journal of Physics 40,1852-1856 (1972) Class project when I taught Jr. CM at Georgia Tech (Just 5 students) From: Geometric aspects of classical Coulomb scattering American Journal of Physics 40,1852-1856 (1972) Class project when I taught Jr. CM at Georgia Tech (Just 5 students) 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 equivalent of Newton’s F=dp/dt equations: N=dL/dt

Angular momentum vector Lj of a mass mj is its linear momentum pj times its lever arm as given by the angular momentum cross-product relation L =r × m r! ≡ r × p j j j j j j Rotational equivalent of Newton’s F=dp/dt equations: N=dL/dt

Angular momentum vector Lj of a mass mj is its linear momentum pj times its lever arm as given by the angular momentum cross-product relation L =r × m r! ≡ r × p j j j j j j 3 3 total The sum-total angular momentum is L = L = ∑ L j= ∑ rj × mjr! j j=1 j=1 Rotational equivalent of Newton’s F=dp/dt equations: N=dL/dt

Angular momentum vector Lj of a mass mj is its linear momentum pj times its lever arm as given by the angular momentum cross-product relation L =r × m r! ≡ r × p j j j j j j 3 3 total The sum-total angular momentum is L = L = ∑ L j= ∑ rj × mjr! j j=1 j=1 dL /dt gives a rotor Newton equation relating rotor momentum rxp to rotor force or torque rxF. 3 3 m dL total 3 = ∑ r × m !r! = ∑ r × F r23 = r2- r3 dt j j j j j j=1 j=1 r3 r31 = r3- r1 3 3 ⎛ 3 ⎞ m applied constraint 2 rCM= m1r1+ m2r2+ m3r3 = ∑ rj × Fj + ∑ rj × ⎜ ∑ Fjk ⎟ m +m +m j 1 j 1 r 1 2 3 = = ⎝ k =1(k ≠ j) ⎠ 2 r12 = r1- r2 m1 r1 Fig. 6.4.1 Three-particle coordinate vectors applied F3

F32 = - F23 m3

F23 = - F32 F31 = - F13

F applied m2 2 F21 = - F12 F13 = - F31

m1 F12 = - F21 applied F1 Fig. 6.4.2 Three-particle force vectors Rotational equivalent of Newton’s F=dp/dt equations: N=dL/dt

Angular momentum vector Lj of a mass mj is its linear momentum pj times its lever arm as given by the angular momentum cross-product relation L =r × m r! ≡ r × p j j j j j j 3 3 total The sum-total angular momentum is L = L = ∑ L j= ∑ rj × mjr! j j=1 j=1 dL /dt gives a rotor Newton equation relating rotor momentum rxp to rotor force or torque rxF. 3 3 m dL total 3 = ∑ r × m !r! = ∑ r × F r23 = r2- r3 dt j j j j j j=1 j=1 r3 r31 = r3- r1 3 3 ⎛ 3 ⎞ m applied constraint 2 rCM= m1r1+ m2r2+ m3r3 = ∑ rj × Fj + ∑ rj × ⎜ ∑ Fjk ⎟ m +m +m j 1 j 1 r 1 2 3 = = ⎝ k =1(k ≠ j) ⎠ 2 r12 = r1- r2 m1 r Internal constraint or coupling force terms appear at first to be a nuisance. 1 Fig. 6.4.1 Three-particle coordinate vectors 3 3 applied constraint constraint constraint constraint F3 ∑ ∑ rj × Fjk = r1 × F12 + F13 + r2 × F21 + F23 + r3 × F31 + F32 j=1 k =1 k ≠ j ( ) ( ) ( ) ( ) F32 = - F23 m3 = r − r × Fconstraint + r − r × Fconstraint + r − r × Fconstraint = 0 ( 1 2 ) 12 ( 1 3 ) 13 ( 2 3 ) 23 F23 = - F32 F31 = - F13

F applied m2 2 F21 = - F12 F13 = - F31

m1 F12 = - F21 applied F1 Fig. 6.4.2 Three-particle force vectors Rotational equivalent of Newton’s F=dp/dt equations: N=dL/dt

Angular momentum vector Lj of a mass mj is its linear momentum pj times its lever arm as given by the angular momentum cross-product relation L =r × m r! ≡ r × p j j j j j j 3 3 total The sum-total angular momentum is L = L = ∑ L j= ∑ rj × mjr! j j=1 j=1 dL /dt gives a rotor Newton equation relating rotor momentum rxp to rotor force or torque rxF. 3 3 m dL total 3 = ∑ r × m !r! = ∑ r × F r23 = r2- r3 dt j j j j j j=1 j=1 r3 r31 = r3- r1 3 3 ⎛ 3 ⎞ m applied constraint 2 rCM= m1r1+ m2r2+ m3r3 = ∑ rj × Fj + ∑ rj × ⎜ ∑ Fjk ⎟ m +m +m j 1 j 1 r 1 2 3 = = ⎝ k =1(k ≠ j) ⎠ 2 r12 = r1- r2 m1 r Internal constraint or coupling force terms appear at first to be a nuisance. 1 Fig. 6.4.1 Three-particle coordinate vectors 3 3 applied constraint constraint constraint constraint F3 ∑ ∑ rj × Fjk = r1 × F12 + F13 + r2 × F21 + F23 + r3 × F31 + F32 j=1 k =1 k ≠ j ( ) ( ) ( ) ( ) F32 = - F23 m3 = r − r × Fconstraint + r − r × Fconstraint + r − r × Fconstraint = 0 ( 1 2 ) 12 ( 1 3 ) 13 ( 2 3 ) 23 F23 = - F32 F31 = - F13

F applied m2 However, they vanish if coupling forces act along lines connecting the masses. 2 F21 = - F12 F13 = - F31

m1 F12 = - F21 applied F1 Fig. 6.4.2 Three-particle force vectors Rotational equivalent of Newton’s F=dp/dt equations: N=dL/dt

Angular momentum vector Lj of a mass mj is its linear momentum pj times its lever arm as given by the angular momentum cross-product relation L =r × m r! ≡ r × p j j j j j j 3 3 total The sum-total angular momentum is L = L = ∑ L j= ∑ rj × mjr! j j=1 j=1 dL /dt gives a rotor Newton equation relating rotor momentum rxp to rotor force or torque rxF. 3 3 m dL total 3 = ∑ r × m !r! = ∑ r × F r23 = r2- r3 dt j j j j j j=1 j=1 r3 r31 = r3- r1 3 3 ⎛ 3 ⎞ m applied constraint 2 rCM= m1r1+ m2r2+ m3r3 = ∑ rj × Fj + ∑ rj × ⎜ ∑ Fjk ⎟ m +m +m j 1 j 1 r 1 2 3 = = ⎝ k =1(k ≠ j) ⎠ 2 r12 = r1- r2 m1 r Internal constraint or coupling force terms appear at first to be a nuisance. 1 Fig. 6.4.1 Three-particle coordinate vectors 3 3 constraint constraint constraint constraint applied F3 ∑ ∑ rj × Fjk = r1 × F12 + F13 + r2 × F21 + F23 + r3 × F31 + F32 j=1k=1(k≠ j) ( ) ( ) ( ) F32 = - F23 m3 = r − r × Fconstraint + r − r × Fconstraint + r − r × Fconstraint = 0 ( 1 2 ) 12 ( 1 3) 13 ( 2 3) 23 F23 = - F32 F31 = - F13

F applied m2 However, they vanish if coupling forces act along lines connecting the masses. 2 F21 = - F12 F13 = - F31

The results are the rotational Newton's equation. m1 F12 = - F21 3 3 applied dL applied F1 = N , where: N = ∑ N and: N = ∑ r × F Fig. 6.4.2 Three-particle force vectors dt j j j j j=1 j=1 Rotational equivalent of Newton’s F=dp/dt equations: N=dL/dt

Angular momentum vector Lj of a mass mj is its linear momentum pj times its lever arm as given by the angular momentum cross-product relation L =r × m r! ≡ r × p j j j j j j 3 3 total The sum-total angular momentum is L = L = ∑ L j= ∑ rj × mjr! j j=1 j=1 dL /dt gives a rotor Newton equation relating rotor momentum rxp to rotor force or torque rxF. 3 3 m dL total 3 = ∑ r × m !r! = ∑ r × F r23 = r2- r3 dt j j j j j j=1 j=1 r3 r31 = r3- r1 3 3 ⎛ 3 ⎞ m applied constraint 2 rCM= m1r1+ m2r2+ m3r3 = ∑ rj × Fj + ∑ rj × ⎜ ∑ Fjk ⎟ m +m +m j 1 j 1 r 1 2 3 = = ⎝ k =1(k ≠ j) ⎠ 2 r12 = r1- r2 m1 r Internal constraint or coupling force terms appear at first to be a nuisance. 1 Fig. 6.4.1 Three-particle coordinate vectors 3 3 constraint constraint constraint constraint applied F3 ∑ ∑ rj × Fjk = r1 × F12 + F13 + r2 × F21 + F23 + r3 × F31 + F32 j=1k=1(k≠ j) ( ) ( ) ( ) F32 = - F23 m3 = r − r × Fconstraint + r − r × Fconstraint + r − r × Fconstraint = 0 ( 1 2 ) 12 ( 1 3) 13 ( 2 3) 23 F23 = - F32 F31 = - F13

F applied m2 However, they vanish if coupling forces act along lines connecting the masses. 2 F21 = - F12 F13 = - F31

The results are the rotational Newton's equation. m1 F12 = - F21 3 3 applied dL applied F1 = N , where: N = ∑ N and: N = ∑ r × F Fig. 6.4.2 Three-particle force vectors dt j j j j j=1 j=1 Taken together with translational Newton's equation the six equations describe rigid body mechanics. 3 dP applied = F , where: F = ∑ F dt j j=1 Rotational equivalent of Newton’s F=dp/dt equations: N=dL/dt

Angular momentum vector Lj of a mass mj is its linear momentum pj times its lever arm as given by the angular momentum cross-product relation L =r × m r! ≡ r × p j j j j j j 3 3 total The sum-total angular momentum is L = L = ∑ L j= ∑ rj × mjr! j j=1 j=1 dL /dt gives a rotor Newton equation relating rotor momentum rxp to rotor force or torque rxF. 3 3 m dL total 3 = ∑ r × m !r! = ∑ r × F r23 = r2- r3 dt j j j j j j=1 j=1 r3 r31 = r3- r1 3 3 ⎛ 3 ⎞ m applied constraint 2 rCM= m1r1+ m2r2+ m3r3 = ∑ rj × Fj + ∑ rj × ⎜ ∑ Fjk ⎟ m +m +m j 1 j 1 r 1 2 3 = = ⎝ k =1(k ≠ j) ⎠ 2 r12 = r1- r2 m1 r Internal constraint or coupling force terms appear at first to be a nuisance. 1 Fig. 6.4.1 Three-particle coordinate vectors 3 3 constraint constraint constraint constraint applied F3 ∑ ∑ rj × Fjk = r1 × F12 + F13 + r2 × F21 + F23 + r3 × F31 + F32 j=1k=1(k≠ j) ( ) ( ) ( ) F32 = - F23 m3 = r − r × Fconstraint + r − r × Fconstraint + r − r × Fconstraint = 0 ( 1 2 ) 12 ( 1 3) 13 ( 2 3) 23 F23 = - F32 F31 = - F13

F applied m2 However, they vanish if coupling forces act along lines connecting the masses. 2 F21 = - F12 F13 = - F31

The results are the rotational Newton's equation. m1 F12 = - F21 3 3 applied dL applied F1 = N , where: N = ∑ N and: N = ∑ r × F Fig. 6.4.2 Three-particle force vectors dt j j j j j=1 j=1 Taken together with translational Newton's equation the six equations describe rigid body mechanics. 3 dP applied = F , where: F = ∑ F dt j j=1 Remaining 3N-6 equations consist of normal mode or GCC equations of some kind. 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 The Australian Boomerang (that comes back!)

dL = N dt

Higher aerodynamic L lift

N

Less aerodynamic lift N Unbalanced lift gives torque N The Australian Boomerang (that comes back and hovers down!)

L dL N = N dt

Higher aerodynamic L lift

N

Less aerodynamic lift N Unbalanced lift gives torque N

Small lifting torque due to “bad-air” of leading blade hitting trailing one left-to-right may cause boomerang to level and hover. Stronger effect in 3-blade boomers causes figure-8 paths. The Australian Boomerang (that comes back and hovers down!)

Charlie Drake’s famous 1961 song: My boomerang won’t come back! My boomerang won’t come back! My boomerang won’t come back! I’ve waved the thing all over the place L N dL Practiced til’ I was blue* in the face = N I’m a big disgrace dt to the Aborigine Race My boomerang won’t come back! Higher aerodynamic L lift

N

*blue later replaced black Less Aluminum boomerang I made in 1965. aerodynamic lift N It once flew over 18 seconds with hover-return! Unbalanced lift gives Song at end of Lect.28 torque N p.90

https://www.youtube.com/playlist?list=PLGwmGldCxzLxbPlFVG8Z89WZIBuT4m0Ii

Small lifting torque due to “bad-air” of leading blade hitting trailing one left-to-right may cause boomerang to level and hover. Stronger effect in 3-blade boomers causes figure-8 paths. 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 The gyrocompass and mechanical spin analogy Suppose Euler ball has right-hand rotation with angular momentum L L The gyrocompass and mechanical spin analogy Suppose Euler ball has right-hand rotation with angular momentum L N If the α-dial for z-rotation is turning left-to-right this applies righthand “thumbs-up” torque N L The gyrocompass and mechanical spin analogy Suppose Euler ball has right-hand rotation with angular momentum L N If the α-dial for z-rotation is turning left-to-right this applies righthand “thumbs-up” torque N L

Then the ball tends to line-up with z-axis (and may go past z, then come back, etc. in a precessional or “hunting” motion) The gyrocompass and mechanical spin analogy Suppose Euler ball has right-hand rotation with angular momentum L N If the α-dial for z-rotation is turning left-to-right this applies righthand “thumbs-up” torque N L

A very high speed ball in a gyro-compass will Then the ball tends to line-up with z-axis similarly seek true North due to Earth rotation. (and may go past z, then come back, etc. in a precessional or “hunting” motion) The gyrocompass and mechanical spin analogy Suppose Euler ball has right-hand rotation with angular momentum L N If the α-dial for z-rotation is turning left-to-right this applies righthand “thumbs-up” torque N L

A very high speed ball in a gyro-compass will Then the ball tends to line-up with z-axis similarly seek true North due to Earth rotation. (and may go past z, then come back, etc. in a precessional or “hunting” motion) This is analogous to the tendency for spin magnetic moments to align (or precess about) the B-direction of a magnetic field Recall S-precession discussion in CMwB Unit 4 Ch.4 and Lect.26 The gyrocompass and mechanical spin analogy Suppose Euler ball has right-hand rotation with angular momentum L N If the α-dial for z-rotation is turning left-to-right this applies righthand “thumbs-up” torque N L

A very high speed ball in a gyro-compass will Then the ball tends to line-up with z-axis similarly seek true North due to Earth rotation. (and may go past z, then come back, etc. in a precessional or “hunting” motion) General Rule: Gyros tend to “line-up” so they are rotating This is analogous to the tendency for spin magnetic moments with whatever is most closely to align (or precess about) the B-direction of a magnetic field coupled to them. Recall S-precession discussion in CMwB Unit 4 Ch.4 and Lect.26 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 Inertia tensors Consider N-body angular velocity ω and angular momentum L relations with Levi-Civita analysis N N r r and L = ∑ r × m r! = ∑ m r × ω × r with A × B × C = A • C B − A • B C ! j = ω × j j j j j j ( j ) ( ) ( ) ( ) j=1 j=1

r = (x , y , z ) = r( 1, 1 ,0) Consider mass m instantaneously at m m m m 2 2 on a bent axle rotating in a fixed bearing: z −ω/2 2 L=mr ω/2 0 x y 0 ω = ω r/√2 m 0 r = r/√2

0 Fig. 6.5.1 Angular momentum for mass rotating on bent axle. Inertia tensors Consider N-body angular velocity ω and angular momentum L relations with Levi-Civita analysis N N and L r m r m r r with A B C A C B A B C r! j = ω × rj = ∑ j × j ! j = ∑ j j × (ω × j ) × ( × ) = ( • ) − ( • ) j=1 j=1 ! N ! N This produces the rotational inertia tensor I: I = ∑ I = ∑ m ⎡ r • r 1− r r ⎤ j j ⎣( j j ) j j ⎦ j=1 j=1 N N ! in the ω-to-L relation: L = ∑ m ⎡ r • r ω − r • ω r ⎤ = ∑ m ⎡ r • r 1− r r ⎤• ω = I • ω j ⎣( j j ) ( j ) j ⎦ j ⎣( j j ) j j ⎦ j=1 j=1

r = (x , y , z ) = r( 1, 1 ,0) Consider mass m instantaneously at m m m m 2 2 on a bent axle rotating in a fixed bearing: z −ω/2 2 L=mr ω/2 0 x y 0 ω = ω r/√2 m 0 r = r/√2

0 Fig. 6.5.1 Angular momentum for mass rotating on bent axle. Inertia tensors Consider N-body angular velocity ω and angular momentum L relations with Levi-Civita analysis N N and L r m r m r r with A B C A C B A B C r! j = ω × rj = ∑ j × j ! j = ∑ j j × (ω × j ) × ( × ) = ( • ) − ( • ) j=1 j=1 ! N ! N This produces the rotational inertia tensor I: I = ∑ I = ∑ m ⎡ r • r 1− r r ⎤ j j ⎣( j j ) j j ⎦ j=1 j=1 N N ! in the ω-to-L relation: L = ∑ m ⎡ r • r ω − r • ω r ⎤ = ∑ m ⎡ r • r 1− r r ⎤• ω = I • ω j ⎣( j j ) ( j ) j ⎦ j ⎣( j j ) j j ⎦ j=1 j=1 Matrix form of the ω-to-L relation using the inertia matrix 〈I〉 ⎛ 2 2 ⎞ ⎛ 2 2 ⎞ ⎛ ⎞ y + z −x y −x z ⎛ ⎞ y + z −x y −x z Lx j j j j j j ω x ⎜ j j j j j j ⎟ ⎜ ⎟ N N ⎜ ⎟ N ⎜ ⎟ ! ! ⎜ 2 2 ⎟ L m ⎜ y x x2 z2 y z ⎟ ω I = ∑ I = ∑ m − y x x + z − y z ⎜ y ⎟ = ∑ j ⎜ − j j j + j − j j ⎟ ⎜ y ⎟ j j ⎜ j j j j j j ⎟ ⎜ ⎟ j=1 ⎜ ⎟ j=1 j=1 ⎜ ⎟ ⎜ 2 2 ⎟ ⎜ L ⎟ 2 2 ⎜ ω ⎟ ⎜ −z x −z y x + y ⎟ ⎝ z ⎠ ⎜ −z jx j −z j y j x j + y j ⎟ ⎝ z ⎠ ⎝ j j j j j j ⎠ ⎝ ⎠

r = (x , y , z ) = r( 1, 1 ,0) Consider mass m instantaneously at m m m m 2 2 on a bent axle rotating in a fixed bearing: z −ω/2 2 L=mr ω/2 0 x y 0 ω = ω r/√2 m 0 r = r/√2

0 Fig. 6.5.1 Angular momentum for mass rotating on bent axle. Inertia tensors Consider N-body angular velocity ω and angular momentum L relations with Levi-Civita analysis N N and L r m r m r r with A B C A C B A B C r! j = ω × rj = ∑ j × j ! j = ∑ j j × (ω × j ) × ( × ) = ( • ) − ( • ) j=1 j=1 ! N ! N This produces the rotational inertia tensor I: I = ∑ I = ∑ m ⎡ r • r 1− r r ⎤ j j ⎣( j j ) j j ⎦ j=1 j=1 N N ! in the ω-to-L relation: L = ∑ m ⎡ r • r ω − r • ω r ⎤ = ∑ m ⎡ r • r 1− r r ⎤• ω = I • ω j ⎣( j j ) ( j ) j ⎦ j ⎣( j j ) j j ⎦ j=1 j=1 Matrix form of the ω-to-L relation using the inertia matrix 〈I〉 ⎛ 2 2 ⎞ ⎛ 2 2 ⎞ ⎛ ⎞ y + z −x y −x z ⎛ ⎞ y + z −x y −x z Lx j j j j j j ω x ⎜ j j j j j j ⎟ ⎜ ⎟ N N ⎜ ⎟ N ⎜ ⎟ ! ! ⎜ 2 2 ⎟ L m ⎜ y x x2 z2 y z ⎟ ω I = ∑ I = ∑ m − y x x + z − y z ⎜ y ⎟ = ∑ j ⎜ − j j j + j − j j ⎟ ⎜ y ⎟ j j ⎜ j j j j j j ⎟ ⎜ ⎟ j=1 ⎜ ⎟ j=1 j=1 ⎜ ⎟ ⎜ 2 2 ⎟ ⎜ L ⎟ 2 2 ⎜ ω ⎟ ⎜ −z x −z y x + y ⎟ ⎝ z ⎠ ⎜ −z jx j −z j y j x j + y j ⎟ ⎝ z ⎠ ⎝ j j j j j j ⎠ ⎝ ⎠

r = (x , y , z ) = r( 1, 1 ,0) Consider mass m instantaneously at m m m m 2 2 on a bent axle rotating in a fixed bearing: z −ω/2 Instantaneous matrix 〈I〉 of inertia is: =mr2 ⎛ 2 ⎞ L ω/2 ⎜ (1/ 2) + 0 −(1/ 2)(1/ 2) −(1/ 2)0 ⎟ 0 ⎜ ⎟ ⎛ 1/2 −1/2 0 ⎞ ! 2 2 2⎜ ⎟ I =mr ⎜ ⎟ =mr x y 0 ⎜ −(1/ 2)(1/ 2) (1/ 2) + 0 −(1/ 2)0 ⎟ ⎜ −1/2 1/2 0 ⎟ ω ⎜ ⎟ ⎜ ⎟ = 2 2 ⎝ 0 0 1 ⎠ ω ⎜ −0 1/ 2 −0 1/ 2 1/ 2 + 1/ 2 ⎟ ⎝⎜ ( ) ( ) ( ) ( ) ⎠⎟ r/√2 m 0 Matrix 〈I〉 operates on angular velocity ω to give angular momentum L r = r/√2

0 Fig. 6.5.1 Angular momentum for mass rotating on bent axle. Inertia tensors Consider N-body angular velocity ω and angular momentum L relations with Levi-Civita analysis N N and L r m r m r r with A B C A C B A B C r! j = ω × rj = ∑ j × j ! j = ∑ j j × (ω × j ) × ( × ) = ( • ) − ( • ) j=1 j=1 ! N ! N This produces the rotational inertia tensor I: I = ∑ I = ∑ m ⎡ r • r 1− r r ⎤ j j ⎣( j j ) j j ⎦ j=1 j=1 N N ! in the ω-to-L relation: L = ∑ m ⎡ r • r ω − r • ω r ⎤ = ∑ m ⎡ r • r 1− r r ⎤• ω = I • ω j ⎣( j j ) ( j ) j ⎦ j ⎣( j j ) j j ⎦ j=1 j=1 Matrix form of the ω-to-L relation using the inertia matrix 〈I〉 ⎛ 2 2 ⎞ ⎛ 2 2 ⎞ ⎛ ⎞ y + z −x y −x z ⎛ ⎞ y + z −x y −x z Lx j j j j j j ω x ⎜ j j j j j j ⎟ ⎜ ⎟ N N ⎜ ⎟ N ⎜ ⎟ ! ! ⎜ 2 2 ⎟ L m ⎜ y x x2 z2 y z ⎟ ω I = ∑ I = ∑ m − y x x + z − y z ⎜ y ⎟ = ∑ j ⎜ − j j j + j − j j ⎟ ⎜ y ⎟ j j ⎜ j j j j j j ⎟ ⎜ ⎟ j=1 ⎜ ⎟ j=1 j=1 ⎜ ⎟ ⎜ 2 2 ⎟ ⎜ L ⎟ 2 2 ⎜ ω ⎟ ⎜ −z x −z y x + y ⎟ ⎝ z ⎠ ⎜ −z jx j −z j y j x j + y j ⎟ ⎝ z ⎠ ⎝ j j j j j j ⎠ ⎝ ⎠

r = (x , y , z ) = r( 1, 1 ,0) Consider mass m instantaneously at m m m m 2 2 on a bent axle rotating in a fixed bearing: z −ω/2 Instantaneous matrix 〈I〉 of inertia is: =mr2 ⎛ 2 ⎞ L ω/2 ⎜ (1/ 2) + 0 −(1/ 2)(1/ 2) −(1/ 2)0 ⎟ 0 ⎜ ⎟ ⎛ 1/2 −1/2 0 ⎞ ! 2 2 2⎜ ⎟ I =mr ⎜ ⎟ =mr x y 0 ⎜ −(1/ 2)(1/ 2) (1/ 2) + 0 −(1/ 2)0 ⎟ ⎜ −1/2 1/2 0 ⎟ ω ⎜ ⎟ ⎜ ⎟ = 2 2 ⎝ 0 0 1 ⎠ ω ⎜ −0 1/ 2 −0 1/ 2 1/ 2 + 1/ 2 ⎟ ⎝⎜ ( ) ( ) ( ) ( ) ⎠⎟ r/√2 m 0 Matrix 〈I〉 operates on angular velocity ω to give angular momentum L r = r/√2 ⎛ ⎞ Lx ⎛ ⎞ ⎛ ⎞ ⎛ ⎞ ⎜ ⎟ 1/2 −1/2 0 0 −1/2 0 L 2 ⎜ ⎟ ⎜ ⎟ 2 ⎜ ⎟ ⎜ y ⎟ = mr ⎜ −1/2 1/2 0 ⎟ ω = mr ⎜ 1/2 ⎟ ω ⎜ ⎟ ⎜ ⎟ Fig. 6.5.1 Angular momentum for mass rotating on bent axle. ⎜ ⎟ ⎝⎜ 0 ⎠⎟ ⎜ ⎟ ⎜ Lz ⎟ ⎝ 0 0 1 ⎠ ⎝ 0 ⎠ ⎝ ⎠ Inertia tensors Consider N-body angular velocity ω and angular momentum L relations with Levi-Civita analysis N N and L r m r m r r with A B C A C B A B C r! j = ω × rj = ∑ j × j ! j = ∑ j j × (ω × j ) × ( × ) = ( • ) − ( • ) j=1 j=1 ! N ! N This produces the rotational inertia tensor I: I = ∑ I = ∑ m ⎡ r • r 1− r r ⎤ j j ⎣( j j ) j j ⎦ j=1 j=1 N N ! in the ω-to-L relation: L = ∑ m ⎡ r • r ω − r • ω r ⎤ = ∑ m ⎡ r • r 1− r r ⎤• ω = I • ω j ⎣( j j ) ( j ) j ⎦ j ⎣( j j ) j j ⎦ j=1 j=1 Matrix form of the ω-to-L relation using the inertia matrix 〈I〉 ⎛ 2 2 ⎞ ⎛ 2 2 ⎞ ⎛ ⎞ y + z −x y −x z ⎛ ⎞ y + z −x y −x z Lx j j j j j j ω x ⎜ j j j j j j ⎟ ⎜ ⎟ N N ⎜ ⎟ N ⎜ ⎟ ! ! ⎜ 2 2 ⎟ L m ⎜ y x x2 z2 y z ⎟ ω I = ∑ I = ∑ m − y x x + z − y z ⎜ y ⎟ = ∑ j ⎜ − j j j + j − j j ⎟ ⎜ y ⎟ j j ⎜ j j j j j j ⎟ ⎜ ⎟ j=1 ⎜ ⎟ j=1 j=1 ⎜ ⎟ ⎜ 2 2 ⎟ ⎜ L ⎟ 2 2 ⎜ ω ⎟ ⎜ −z x −z y x + y ⎟ ⎝ z ⎠ ⎜ −z jx j −z j y j x j + y j ⎟ ⎝ z ⎠ ⎝ j j j j j j ⎠ ⎝ ⎠

r = (x , y , z ) = r( 1, 1 ,0) Consider mass m instantaneously at m m m m 2 2 on a bent axle rotating in a fixed bearing: z −ω/2 Instantaneous matrix 〈I〉 of inertia is: =mr2 ⎛ 2 ⎞ L ω/2 ⎜ (1/ 2) + 0 −(1/ 2)(1/ 2) −(1/ 2)0 ⎟ 0 ⎜ ⎟ ⎛ 1/2 −1/2 0 ⎞ ! 2 2 2⎜ ⎟ I =mr ⎜ ⎟ =mr x y 0 ⎜ −(1/ 2)(1/ 2) (1/ 2) + 0 −(1/ 2)0 ⎟ ⎜ −1/2 1/2 0 ⎟ ω ⎜ ⎟ ⎜ ⎟ = 2 2 ⎝ 0 0 1 ⎠ ω ⎜ −0 1/ 2 −0 1/ 2 1/ 2 + 1/ 2 ⎟ ⎝⎜ ( ) ( ) ( ) ( ) ⎠⎟ r/√2 m 0 Matrix 〈I〉 operates on angular velocity ω to give angular momentum L r = r/√2 ⎛ L ⎞ dL x ⎛ 1/2 −1/2 0 ⎞ ⎛ ⎞ ⎛ −1/2 ⎞ N L ⎜ ⎟ 0 0 Bearing torque is: = =ω × L 2 ⎜ ⎟ ⎜ ⎟ 2 ⎜ ⎟ dt ⎜ y ⎟ = mr ⎜ −1/2 1/2 0 ⎟ ω = mr ⎜ 1/2 ⎟ ω ⎜ ⎟ ⎜ ⎟ Fig. 6.5.1 Angular momentum for mass rotating on bent axle. ⎜ ⎟ ⎝⎜ 0 ⎠⎟ ⎜ ⎟ ⎜ Lz ⎟ ⎝ 0 0 1 ⎠ ⎝ 0 ⎠ ⎝ ⎠ Kinetic energy in terms of velocity ω and rotational Lagrangian Kinetic energy T of a rotating rigid body can be expressed in terms of the inertia matrix I 1 3 1 3 T = ∑ m r! • r! = ∑ m ω × r • ω × r 2 j j j 2 j ( j ) ( j ) Levi-Civita identity j=1 j=1 3 A × B × C × D = A • C B • D − A • D B • C 1 ⎡ ⎤ ( ) ( ) ( )( ) ( )( ) T = ∑ mj (ω • ω) rj • rj − ω • rj rj • ω 2 j=1 ⎣ ( ) ( )( )⎦ 3 1 ⎡ ⎤ = ω • ∑ mj rj • rj 1 − rj rj • ω 2 j=1 ⎣( ) ( )( )⎦ 1 ! = ω • I • ω 2 Kinetic energy is a quadratic form ⎛ I I I ⎞ ⎛ ⎞ xx xy xz ω x 1 ⎜ ⎟ ⎜ ⎟ T = ω x ω y ω y ⎜ I yx I yy I yz ⎟ ⎜ ω y ⎟ 2 ( )⎜ ⎟ ⎜ ⎟ ⎜ I I I ⎟ ⎜ ω ⎟ ⎝ zx zy zz ⎠ ⎝ z ⎠ ⎛ x I x x I y x I z ⎞ ⎛ x ω ⎞ 1 ⎜ ⎟ ⎜ ⎟ = ω x ω y ω z ⎜ y I x y I y y I z ⎟ ⎜ y ω ⎟ 2 ( )⎜ ⎟ ⎜ ⎟ (Dirac notation) ⎜ z I x z I y z I z ⎟ ⎜ z ⎟ ⎝ ⎠ ⎝ ω ⎠

⎛ 2 2 ⎞ y + z −x y −x z ⎛ ⎞ ⎜ j j j j j j ⎟ ω x 1 3 ⎜ ⎟ ω ω ω m ⎜ y x x2 z2 y z ⎟ ω = x y y ∑ j ⎜ − j j j + j − j j ⎟ ⎜ y ⎟ 2 ( ) j=1 ⎜ ⎟ ⎜ 2 2 ⎟ ⎜ ω ⎟ ⎜ −z j x j −z j y j x j + y j ⎟ ⎝ z ⎠ ⎝ ⎠ Simplifies in principle inertial axes {X,Y,Z}or body eigen-axes ⎛ ⎞ ⎛ ⎞ I XX I XY I XZ ω X 1 ⎜ ⎟ ⎜ ⎟ T = ω X ωY ωZ ⎜ IYX IYY IYZ ⎟ ⎜ ωY ⎟ 2 ( )⎜ ⎟ ⎜ ⎟ I I I ⎝⎜ ZX ZY ZZ ⎠⎟ ⎝⎜ ωZ ⎠⎟ ⎛ I 0 0 ⎞ ⎛ ω ⎞ ⎜ XX ⎟ ⎜ X ⎟ 2 2 2 1 I XXω X IYYωY IZZωZ = ω X ωY ωZ ⎜ 0 IYY 0 ⎟ ⎜ ωY ⎟ = + + 2 ( )⎜ ⎟ ⎜ ⎟ 2 2 2 0 0 I ⎝⎜ ZZ ⎠⎟ ⎝⎜ ωZ ⎠⎟

Kinetic energy in terms of momentum L and rotational Hamiltonian ! ! L = I • ω , generally implies: ω = I−1 • L

Express kinetic energy T in terms of angular velocity ω , momentum L, or both at once. once 1 ! 1 1 1 ! T = ω • I • ω = ω • L= L • ω = L • I−1 • L 2 2 2 2 −1 ⎛ ⎞ ⎛ L ⎞ I XX I XY I XZ X 1 ⎜ ⎟ ⎜ ⎟ T L L L ⎜ ⎟ L = X Y Z IYX IYY IYZ ⎜ Y ⎟ 2 ( )⎜ ⎟ ⎜ ⎟ I I I ⎜ L ⎟ ⎝⎜ ZX ZY ZZ ⎠⎟ ⎝ Z ⎠ ⎛ 1/ I 0 0 ⎞ ⎛ L ⎞ ⎜ XX ⎟ ⎜ X ⎟ 2 2 2 1 LX LY LZ = LX LY LZ ⎜ 0 1/ IYY 0 ⎟ ⎜ LY ⎟ = + + 2 ( )⎜ ⎟ ⎜ ⎟ 2I XX 2IYY 2IZZ 0 0 1/ I L ⎝⎜ ZZ ⎠⎟ ⎝⎜ Z ⎠⎟

Kinetic energy in terms of momentum L and rotational Hamiltonian ! ! L = I • ω , generally implies: ω = I−1 • L

Express kinetic energy T in terms of angular velocity ω , momentum L, or both at once. once 1 ! 1 1 1 ! T = ω • I • ω = ω • L= L • ω = L • I−1 • L 2 2 2 2 −1 ⎛ ⎞ ⎛ L ⎞ I XX I XY I XZ X 1 ⎜ ⎟ ⎜ ⎟ T L L L ⎜ ⎟ L = X Y Z IYX IYY IYZ ⎜ Y ⎟ 2 ( )⎜ ⎟ ⎜ ⎟ I I I ⎜ L ⎟ ⎝⎜ ZX ZY ZZ ⎠⎟ ⎝ Z ⎠ ⎛ 1/ I 0 0 ⎞ ⎛ L ⎞ ⎜ XX ⎟ ⎜ X ⎟ 2 2 2 1 LX LY LZ = LX LY LZ ⎜ 0 1/ IYY 0 ⎟ ⎜ LY ⎟ = + + 2 ( )⎜ ⎟ ⎜ ⎟ 2I XX 2IYY 2IZZ ⎜ 0 0 1/ I ⎟ ⎜ L ⎟ ⎝ ZZ ⎠ ⎝ Z ⎠ L Hamiltonian form is the equation of the angular momentum or L-ellipsoid Lagrangian form is the equation of the angular velocity or ω-ellipsoid Kinetic energy in terms of momentum L and rotational Hamiltonian ! ! L = I • ω , generally implies: ω = I−1 • L

Express kinetic energy T in terms of angular velocity ω , momentum L, or both at once. once 1 ! 1 1 1 ! T = ω • I • ω = ω • L= L • ω = L • I−1 • L 2 2 2 2 −1 ⎛ ⎞ ⎛ L ⎞ I XX I XY I XZ X 1 ⎜ ⎟ ⎜ ⎟ T L L L ⎜ ⎟ L = X Y Z IYX IYY IYZ ⎜ Y ⎟ 2 ( )⎜ ⎟ ⎜ ⎟ I I I ⎜ L ⎟ ⎝⎜ ZX ZY ZZ ⎠⎟ ⎝ Z ⎠ ⎛ 1/ I 0 0 ⎞ ⎛ L ⎞ ⎜ XX ⎟ ⎜ X ⎟ 2 2 2 1 LX LY LZ = LX LY LZ ⎜ 0 1/ IYY 0 ⎟ ⎜ LY ⎟ = + + 2 ( )⎜ ⎟ ⎜ ⎟ 2I XX 2IYY 2IZZ ⎜ 0 0 1/ I ⎟ ⎜ L ⎟ ⎝ ZZ ⎠ ⎝ Z ⎠ L Hamiltonian form is the equation of the angular momentum or L-ellipsoid Lagrangian form is the equation of the angular velocity or ω-ellipsoid

Recall quadratic forms for Lagrangian and Hamiltonian in Lecture 8 and CMwBANG! Unit 1 Ch.12 Unit 1 Fig. 12.2 p. 25 of Lagrangian plot Hamiltonian plot (a) -1 Lecture 8 L(v)=const.=v•M•v/2 (b)H(p)=const.=p•M •p/2 p2=m2v2 H=const = E

p p1 v =p /m 2 2 2 =m1v1

L=const = E

(c) Overlapping plots Lagrangian tangent at velocity v is normal to momentum p 1st equation of Lagrange v L=const = E p = ∇vL =M•v st v v = H v = 1 equation of Hamilton ∇p 1 -1 H=const = E =M •p p1 /m1 p

v (d) Less mass Hamiltonian tangent at momentum p

p v is normal to velocity v

p

(e) More mass Kinetic energy in terms of momentum L and rotational Hamiltonian ! ! L = I • ω , generally implies: ω = I−1 • L

Express kinetic energy T in terms of angular velocity ω , momentum L, or both at once. once 1 ! 1 1 1 ! Torque-free body T = ω • I • ω = ω • L= L • ω = L • I−1 • L has conserved L=const. 2 2 2 2 −1 ⎛ ⎞ ⎛ L ⎞ I XX I XY I XZ X 1 ⎜ ⎟ ⎜ ⎟ T L L L ⎜ ⎟ L = X Y Z IYX IYY IYZ ⎜ Y ⎟ 2 ( )⎜ ⎟ ⎜ ⎟ I I I ⎜ L ⎟ ⎝⎜ ZX ZY ZZ ⎠⎟ ⎝ Z ⎠ ⎛ 1/ I 0 0 ⎞ ⎛ L ⎞ ⎜ XX ⎟ ⎜ X ⎟ 2 2 2 1 LX LY LZ = LX LY LZ ⎜ 0 1/ IYY 0 ⎟ ⎜ LY ⎟ = + + 2 ( )⎜ ⎟ ⎜ ⎟ 2I XX 2IYY 2IZZ ⎜ 0 0 1/ I ⎟ ⎜ L ⎟ ⎝ ZZ ⎠ ⎝ Z ⎠ L = 2T if energy Hamiltonian form is the equation of the angular momentum or L-ellipsoid is not dissipated internally Lagrangian form is the equation of the angular velocity or ω-ellipsoid ω is generally not conserved unless it is aligned to L or body has symmetry Kinetic energy in terms of momentum L and rotational Hamiltonian ! ! L = I • ω , generally implies: ω = I−1 • L

Express kinetic energy T in terms of angular velocity ω , momentum L, or both at once. once 1 ! 1 1 1 ! Torque-free body T = ω • I • ω = ω • L= L • ω = L • I−1 • L has conserved L=const. 2 2 2 2 −1 ⎛ ⎞ ⎛ L ⎞ I XX I XY I XZ X 1 ⎜ ⎟ ⎜ ⎟ T L L L ⎜ ⎟ L = X Y Z IYX IYY IYZ ⎜ Y ⎟ 2 ( )⎜ ⎟ ⎜ ⎟ I I I ⎜ L ⎟ ⎝⎜ ZX ZY ZZ ⎠⎟ ⎝ Z ⎠ ⎛ 1/ I 0 0 ⎞ ⎛ L ⎞ ⎜ XX ⎟ ⎜ X ⎟ 2 2 2 1 LX LY LZ = LX LY LZ ⎜ 0 1/ IYY 0 ⎟ ⎜ LY ⎟ = + + 2 ( )⎜ ⎟ ⎜ ⎟ 2I XX 2IYY 2IZZ ⎜ 0 0 1/ I ⎟ ⎜ L ⎟ ⎝ ZZ ⎠ ⎝ Z ⎠ L = 2T if energy Hamiltonian form is the equation of the angular momentum or L-ellipsoid is not dissipated internally Lagrangian form is the equation of the angular velocity or ω-ellipsoid ω is generally not conserved unless it is aligned to L or body has symmetry

∂L Canonical momentum: pµ = (where: L = T ) ∂q!µ ∂T ∂ ω • Ι • ω L = = ∇ωT = = Ι • ω ∂ω ∂ω 2 Kinetic energy in terms of momentum L and rotational Hamiltonian ! ! L = I • ω , generally implies: ω = I−1 • L

Express kinetic energy T in terms of angular velocity ω , momentum L, or both at once. once 1 ! 1 1 1 ! Torque-free body T = ω • I • ω = ω • L= L • ω = L • I−1 • L has conserved L=const. 2 2 2 2 −1 ⎛ ⎞ ⎛ L ⎞ I XX I XY I XZ X 1 ⎜ ⎟ ⎜ ⎟ T L L L ⎜ ⎟ L = X Y Z IYX IYY IYZ ⎜ Y ⎟ 2 ( )⎜ ⎟ ⎜ ⎟ I I I ⎜ L ⎟ ⎝⎜ ZX ZY ZZ ⎠⎟ ⎝ Z ⎠ ⎛ 1/ I 0 0 ⎞ ⎛ L ⎞ ⎜ XX ⎟ ⎜ X ⎟ 2 2 2 1 LX LY LZ = LX LY LZ ⎜ 0 1/ IYY 0 ⎟ ⎜ LY ⎟ = + + 2 ( )⎜ ⎟ ⎜ ⎟ 2I XX 2IYY 2IZZ ⎜ 0 0 1/ I ⎟ ⎜ L ⎟ ⎝ ZZ ⎠ ⎝ Z ⎠ L = 2T if energy Hamiltonian form is the equation of the angular momentum or L-ellipsoid is not dissipated internally Lagrangian form is the equation of the angular velocity or ω-ellipsoid ω is generally not conserved unless it is aligned to L or body has symmetry

∂L Canonical momentum: pµ = (where: L = T ) ∂q!µ ∂T ∂ ω • Ι • ω L = = ∇ωT = = Ι • ω ∂ω ∂ω 2

∂H Hamilton's 1st equations : q!µ = (where: H = T ) ∂p −1 µ ∂H ∂ L • Ι • L −1 ω = = ∇L H = = Ι • L ∂L ∂L 2 Kinetic energy in terms of momentum L and rotational Hamiltonian ! ! L = I • ω , generally implies: ω = I−1 • L

Express kinetic energy T in terms of angular velocity ω , momentum L, or both at once. once 1 ! 1 1 1 ! Torque-free body T = ω • I • ω = ω • L= L • ω = L • I−1 • L has conserved L=const. 2 2 2 2 −1 ⎛ ⎞ ⎛ L ⎞ I XX I XY I XZ X 1 ⎜ ⎟ ⎜ ⎟ T L L L ⎜ ⎟ L = X Y Z IYX IYY IYZ ⎜ Y ⎟ 2 ( )⎜ ⎟ ⎜ ⎟ I I I ⎜ L ⎟ ⎝⎜ ZX ZY ZZ ⎠⎟ ⎝ Z ⎠ ⎛ 1/ I 0 0 ⎞ ⎛ L ⎞ ⎜ XX ⎟ ⎜ X ⎟ 2 2 2 1 LX LY LZ = LX LY LZ ⎜ 0 1/ IYY 0 ⎟ ⎜ LY ⎟ = + + 2 ( )⎜ ⎟ ⎜ ⎟ 2I XX 2IYY 2IZZ ⎜ 0 0 1/ I ⎟ ⎜ L ⎟ ⎝ ZZ ⎠ ⎝ Z ⎠ L = 2T if energy Hamiltonian form is the equation of the angular momentum or L-ellipsoid is not dissipated internally Lagrangian form is the equation of the angular velocity or ω-ellipsoid ω is generally not conserved unless it is aligned to L or body has symmetry

∂L L Canonical momentum: pµ = (where: L = T ) ∂q!µ ∂T ∂ ω • Ι • ω L = = ∇ωT = = Ι • ω ∂ω ∂ω 2

∂H Hamilton's 1st equations : q!µ = (where: H = T ) ∂p −1 µ ∂H ∂ L • Ι • L −1 ω = = ∇L H = = Ι • L ∂L ∂L 2 Kinetic energy in terms of momentum L and rotational Hamiltonian ! ! L = I • ω , generally implies: ω = I−1 • L

Express kinetic energy T in terms of angular velocity ω , momentum L, or both at once. once 1 ! 1 1 1 ! Torque-free body T = ω • I • ω = ω • L= L • ω = L • I−1 • L has conserved L=const. 2 2 2 2 −1 ⎛ ⎞ ⎛ L ⎞ I XX I XY I XZ X 1 ⎜ ⎟ ⎜ ⎟ T L L L ⎜ ⎟ L = X Y Z IYX IYY IYZ ⎜ Y ⎟ 2 ( )⎜ ⎟ ⎜ ⎟ I I I ⎜ L ⎟ ⎝⎜ ZX ZY ZZ ⎠⎟ ⎝ Z ⎠ ⎛ 1/ I 0 0 ⎞ ⎛ L ⎞ ⎜ XX ⎟ ⎜ X ⎟ 2 2 2 1 LX LY LZ = LX LY LZ ⎜ 0 1/ IYY 0 ⎟ ⎜ LY ⎟ = + + 2 ( )⎜ ⎟ ⎜ ⎟ 2I XX 2IYY 2IZZ ⎜ 0 0 1/ I ⎟ ⎜ L ⎟ ⎝ ZZ ⎠ ⎝ Z ⎠ L = 2T if energy Hamiltonian form is the equation of the angular momentum or L-ellipsoid is not dissipated internally Lagrangian form is the equation of the angular velocity or ω-ellipsoid ω is generally not conserved unless it is aligned to L or body has symmetry

∂L L Canonical momentum: pµ = (where: L = T ) ∂q!µ ∂T ∂ ω • Ι • ω L = = ∇ωT = = Ι • ω ∂ω ∂ω 2

∂H Hamilton's 1st equations : q!µ = (where: H = T ) ∂p −1 µ ∂H ∂ L • Ι • L −1 ω = = ∇L H = = Ι • L ∂L ∂L 2

In body frame momentum L moves along intersection of L-ellipsoid and L-sphere (Length |L| is constant in any classical frame.) Asymmetric Top Demo video

Rotational Energy Surfaces (RES) Symmetric, asymmetric, and spherical-top dynamics (Constant L) BOD-frame cone rolling on LAB frame cone

Singular Motion of Asymetric Rotators AJP 44, 11 p1080 1976 Asymmetric-top dynamics (Constant L)

1. NASA Space station video 2. UAF lab air-supported asymmetric top video

https://youtu.be/HWjGvCaqx5g https://youtu.be/1n-HMSCDYtM For those physist who are brave of heart, make note the video’s comments 3. NASA-Rotating Solid Bodies in Microgravity (2008) 4. Early NASA-JPL satellite blunder (1958)

To be Continued ⇒several pages ahead

https://www.youtube.com/watch?v=BPMjcN-sBJ4

Singular Motion of Asymetric Rotators AJP 44, 11 p1080 1976 Comments following Space Lab video of asymmetric rotation show that it is not a widely understood phenomenon Bocce-Ball Asymmetric Top we built at USC (donated to Cal. Museum of Science & Industry)

Asym. Rotor AJP 44,11 1976

Singular Motion of Asymetric Rotators AJP 44, 11 p1080 1976 Bocce-Ball Asymmetric Top Motion solved by Euler’s equation and elliptic integrals

Singular Motion of Asymetric Rotators AJP 44, 11 p1080 1976 Bocce-Ball Asymmetric Top Motion solved by Euler’s equation and elliptic integrals

Singular Motion of Asymetric Rotators AJP 44, 11 p1080 1976 Bocce-Ball Asymmetric Top Motion solved by Euler’s equation and elliptic integrals Bocce-Ball Asymmetric Top Motion solved by Euler’s equation and elliptic integrals 4. Early NASA-JPL satellite blunder (1958)

From text*in preparation by Rick Heller on semi- classical dynamics of polyatomic molecules *Now available: Eric Heller, The Semiclassical Way Princeton University Press (2018) 1 I Rotational Energy Surfaces (RES) and Constant Energy Surfaces (CES) replace Lagrange Poinsot 2ωi iω

Rotational Energy Surface (RES) is* Constant Energy Surface (CES) is quadratic multipole function plotted radially asymmetric ellipsoid of constant E

2 2 2 2 2 2 J Jy Jz Jx Jy Jz Here notation L or L E = x + + with J = const. E = + + = const. 2I 2I 2I 2I x 2I y 2Iz x y z for angular momentum 2 2 2 2 2 2 J2 2 ⎛ sin θ cos φ sin θ sin φ cos θ ⎞ Jx y Jz is replaced by J or J = J 2 + + or : + + = 1 ⎜ ⎟ 2EI 2EI 2EI ⎝ 2I x 2I y 2Iz ⎠ x y z

(a) RE surface x3- (b) CE surface (c) RES intersecting CES _ _ r =√7 _ 3 I =6 I_ =4 I_ =3 J J r =√5 r =√1 3 2 1 1 2 3 3 _ r r1 2 x2- J x1- 3 r3

_ J2 _ J _ J _ 1 J1 2 E = const. J = const.

* First published in: J. Chem. Phys. 80, 4241 (1984) W. G. Harter and C. W. Patterson RES and CES for nearly-symmetric prolate rotors and nearly-symmetric oblate rotors (a) I_ =5.6 _ _ 2 (b) I =5.0 (c) I =3.2 I_ =6 I_ =3 2 2 1 3 γ =75° γ =21° B γB=63° B

- J1

J- - 2 J2

CES

RES RES

- J2- - J2 J- 2 J1 1 nearly-prolate nearly-oblate symmetric rotor asymmetric rotor symmetric rotor RES RES RES RES for symmetric prolate rotor locates J =10 quantum (-J

Minimum uncertainty angle 10 Polar J=10 Θ+10=17.55° cone K=+10 Uncertainty prolate 10 symmetric top +9 Θ+9 angles RES contour K=+ J K RES 10 Θ =cos-1 +8 Θ+8 K √J(J+1) 10 + +7 Θ √J(J+1) ~J+1/2 +7 +6 10 Θ+6 10.488~10.5 + +5 10 Θ+5 + +4 10 Θ+4 +3 10 K=5 + Θ+3 + +2 Θ10 + +2 + +1 10 =84.53° + Θ+1 + 0 - - - -1 J =4 Quantum cones - - -2 - -3 - -4 -5 - -6 - -7 -8 - -9 -10 Link to pdf of: W. G. Harter and J C. Mitchell ,International Journal of Molecular Science, 14, 714-806 (2013) Fig. 1-5 p.730 RES for symmetric and asymmetric rotor approximates J =10 (-J

Minimum uncertainty angle 10 Polar Separatrix circle pair J=10 Θ+10=17.55° RES contours cone K=+10 Uncertainty dihedral angle prolate 10 K ~10 symmetric top +9 Θ+9 angles C RES contour K=+ J K RES 10 Θ =cos-1 A-B +8 Θ+8 K √J(J+1) θ =atan( ) 10 9 sep + +7 Θ B-C √J(J+1) ~J+1/2 +7 +6 10 Θ+6 8 10.488~10.5 + +5 10 Θ+5 π - θ + +4 10 7 sep Θ+4 +3 10 K=5 + Θ+3 6 + +2 Θ10 + +2 + +1 Θ10 =84.53° 6 K ~10 θsep + +1 7 8 9 KA~10 + 0 - - - -1 - - -2 - -3 -6 -4 - -7 -5 - -6 -8 - -7 -8 - -9 -9 -10 -10

Link to pdf of: W. G. Harter and J C. Mitchell ,International Journal80 of Molecular Science, 14, 714-806 (2013) Fig. 1-5 p.730 RES for symmetric prolate rotor locates J =10 quantum (-J

Link to pdf of: W. G. Harter and J C. Mitchell ,International Journal of Molecular Science, 14, 714-806 (2013) Fig. 1-5 p.730 Rotational Energy Surfaces (RES) Symmetric, asymmetric, and spherical-top dynamics (Constant J) BOD-frame cone rolling on LAB frame cone RES for spherical rotor approximates J =88 (-J

Link to pdf of: W. G. Harter and J C. Mitchell ,International Symposium on Molecular Spectroscopy, OSU Columbus (2009) Link to pdf of: W. G. Harter and J C. Mitchell ,International Symposium on Molecular Spectroscopy, OSU Columbus (2009) Rotational Energy Surfaces (RES) Symmetric, asymmetric, and spherical-top dynamics (Constant J) BOD-frame cone rolling on LAB frame cone (a) Constrained rotor (b) Angular velocity ω (c) Energy ellipsoids x LAB and momentum J x3 3 J x axis β BOD 3 x1 x3 axis J ω J-ellipsoid ω x2 β r α LAB ω-ellipsoid x1 axis Fig. 6.7.1 Elementary ω-constrained rotor and angular velocity-momentum geometry.

(a) Constrained rotor:LAB-fixed ω, moving J (b) Free rotor:LAB-fixed J, moving ω β LAB β BOD LAB-fixed x3 axis BOD x3 axis J J x3 axis J(t) ω LAB-fixed ω ω(0) LAB-moving LAB-moving ω(t) α LAB α LAB x1 axis x1 axis

Fig. 6.7.2 Free rotor cut loose from LAB-constraining ω-axis changes dynamics accordingly.

..this was the kind of dynamics that started me dropping superballs... Prolate tops: (a) III=4I3 (b) III=2I3 (c) III=(3/2) I3 • • γ=3αcosβ γ•= α• cosβ γ•=(1/2)α• cosβ γ•=(3/4)ω- γ•=(1/2)ω- γ•=(1/3)ω- 3 3 β 3 LAB x3 - β BOD x ω1 ω axis • • 3 ω- α+ γ=ω axis 1 ω3 ω LAB • BOD ω- α cone 3 cone • • • •- α γ =γx γ• β γ 3 (d) Spherical top: β III= I3 (e) Oblate limit: • β ω γ=0 III=(1/2) I3 α• α• =ω • • γ•=(-1/2)α•cosβ α=αx • 3 • - γ γ•=0 γ= -ω3 Blue BOD-frame cones roll (around ω-sticking axis)without slipping on red LAB-frame cone I −I Fig. 6.7.3 Symmetric top ω-cones for β=30°and inertial ratios: (a) I I 3 =3, (b) 1, (c) 1 ,(d) 0, (e) - 1 . I3 2 2 • ω =γ sin β LAB x 1 BOD x- axis axis 3 3 • • • (a) Prolate geometry ω- =−αsin β α+ γcos β=ω • 1 3 α+γ•=ω III=I1=I2=(3/2) I3 • α= J3/I1= J/I1 β • - αcos β= J3 /I1 - • • =ω-I /I ω3 =αcos β+ γ 3 3 1 • =α(I /I )cos β ω•I•ω=2E • 1 3 prolate α ellipsoid - BOD −x1 axis

β

• • - • γ = ω3 - α cos β γ • β = (α cos β)(I1-I3)/I3 = ω- (I -I )/I LAB −x1 axis 3 1 3 1 Blue BOD-frame cones α• roll without slipping (b) Oblate geometry on red LAB-frame cone III=I1=I2=(1/2) I3 - • x3 α+γ•=ω β

β

ω•I•ω=2E oblate ellipsoid γ•

Fig. 6.7.4 Detailed geometry of symmetric top kinetics. (a) Prolate case. (b) Most-oblate case • - • Oblate limit: γ = ω3 - α cos β Very prolate top: I =9I • II 3 = (α cos β)(I -I )/I III=(1/2) I3 1 3 3 • • = ω- (I -I )/I γ=8αcosβ •=(-1/2) •cos 3 1 3 1 γ α β • - • γ=(8/9)ω3 γ= -ω LAB x3 3 BOD x- axis α• + γ• =ω 3 β axis BOD x- LAB 3 cone ω axis β=30° BOD α• cone • γ Blue BOD-frame cones • • α=αx3 roll without slipping

LAB x3 on red LAB-frame cone β=60° axis LAB cone ω - BOD x3 • axis • • α α+ γ=ω - BOD x3 • β=60° axis γ BOD • • cone α=αx3 Fig. 6.7.5 Extreme cases (Oblate vs. Prolate) of symmetric-top geometry. The following treatment of lever rotation was covered in Lecture 18 (2019)

Cycloidal geometry of flying levers Practical poolhall application (See above link)

Now enjoy this famous hit physics song…

Aluminum boomerang from p.43

https://www.youtube.com/playlist?list=PLGwmGldCxzLxbPlFVG8Z89WZIBuT4m0Ii

90 The Zamboni-Ice-Shot problem

(Assumes frictionless ice rink)

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1 9 0 0 0 10 9 0 0 d= _157_ cm Marbles: A B C Where on a meter-stick do you hit it so as to not disturb marbles A or B and...... knock marble C down as shown.

91