1. PHY 711 Classical Mechanics and Mathematical Methods
11-11:50 AM MWF Olin 102
Plan for Lecture 38
General review
Classical mechanics – traditional topics
Classical mechanics – related topics
Mathematical methods
Course assessment
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PHY 711 Fall Lecture 38

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PHY 711 Fall Lecture 34
PHY 711 Fall Lecture 38 2

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5. Review of course topics
Classical mechanics – traditional topics
Particle scattering; scattering cross section
Inertial and accelerated reference frames
Calculus of variations
Lagrangian and Hamiltonian formalisms
Small oscillations near equilibrium
Rigid body rotations
Classical mechanics – additional topics
Mechanics of continuous media
Mechanics of fluids; bulk and surface motions
Heat conduction
Elastic continua
Mathematical methods
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PHY 711 Fall Lecture 38

6. Scattering theory:
detector
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PHY 711 Fall Lecture 38

7. If we can assume that the process is isotropic in f; results depend on b and q:
Figure from Marion & Thorton, Classical Dynamics
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PHY 711 Fall Lecture 38

8. In general, scattering is most easily analyzed in the center of mass frame -- relationship between center of mass and laboratory frames of reference: v1 V1
VCM y q
lab CM
for elastic case
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9. Differential cross sections in different reference frames
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10. Differential cross sections in different reference frames –
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12. Physics of rotating coordinate system
Effects on acceleration (rotation only):
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13. Equation of motion on Earth’s surfacew
y (east)
z (up) q
x (south)
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14. Calculus of variation example for a pure integral functions
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15. Lamp shade shape y(x) y xi yi x xf yf 12/05/2016
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18. Application of calculus of variation to Lagrangian mechanics:
Potential energy
Kinetic energy
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20. Recipe for constructing the Hamiltonian and analyzing the equations of motion
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PHY 711 Fall Lecture 38

21. Use of Lagrangian/Hamiltonian formalism for molecular dynamics
rij
Pairwise interaction between particles
From this Lagrangian, can find the 3N coupled 2nd order differential equations of motion and/or find the corresponding Hamiltonian, representing the system at constant energy, volume, and particle number N (N,V,E ensemble).
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PHY 711 Fall Lecture 38

22. H. C. Andersen’s effective Lagrangian for constant pressure
simulations
Equations of motion in “original” coordinates:
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PHY 711 Fall Lecture 38

23. Normal modes of vibration about equilibrium:
Example – linear molecule m1 m2 m3
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24. m1 m2 m3 m1 m2 m3 m1 m2 m3
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25. Consider an extended system of masses and springs:
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27. 3-dimensional periodic lattices
Extension of ideas --
3-dimensional periodic lattices
Example – face-centered-cubic unit cell (Al or Ni)
Diagram of atom positions
Diagram of q-space n(q)
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28. From: PRB 59 3395 (1999); Mishin et. al. n(q)
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29. Euler angles for expressing rigid body rotation:
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30. Rotational kinetic energy for “symmetric” top:
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31. Equations of motion for continuous media
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32. Stress tensor for fluids
Stress tensor for elastic media (Hooke’s law in terms of displacement u; linear terms only )
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PHY 711 Fall Lecture 38