Classical Mechanics (P601), Fall 2023

Classical Mechanics (P601), Fall 2023

Instructor: Romain Vasseur, Associate Professor
office: Hasbrouck 405A
email: rvasseur[at]umass[dot]edu
office hours: zoom, office visits and email, preferred times right before or after class.

Lectures: MoWeFr 10.10-11am, Has 130. 

Suggested reading materials:

The lectures and lecture notes are designed to be self-contained, and no textbook is required. If you want to read a book, here are some classics:

Website: https://websites.umass.edu/rvasseur/teaching/. Course materials will also be made available on Slack.

Grading:
The course grade will be based 35% on the homework, 30% on the midterm, and 35% on the final exam. Late homework will receive 50% credit if returned before the solutions are posted, one exception allowed per semester only. 

Topics:

  • Introduction: Newtonian mechanics, conservative forces in 1d.
  • Variational calculus: functionals, Euler-Lagrange equations and constraints.
  • Lagrangian mechanics: Simple examples, symmetries and conservation laws, constraints, motion of a charged particle. 
  • Two-body problems: Reduced coordinates, effective potential, Kepler’s problem, integral solution. 
  • Small oscillations: Stability, damped oscillations, forced oscillations, Green’s functions, coupled oscillations, normal modes.
  • Hamiltonian mechanics: Hamilton’s equations, Liouville’s theorem, Poisson brackets and canonical transformations, symplectic structure. Action-angle variables and integrable systems. Hamilton-Jacobi equation. 
  • Rigid bodies: Kinematics, inertia tensor, Euler’s equations, free tops, Euler’s angles. 

Lecture notes:

  1. Newtonian mechanics
  2. Variational calculus 
  3. Lagrangian mechanics
  4. Two-body problem
  5. Small oscillations
  6. Hamiltonian mechanics
  7. Rigid bodies
  8. Classical field theory

Additional Latex notes on Noether’s theorem, and on the Kepler problem. 

Problem sets: see Slack for problem sets and solutions

Schedule:
9/8: Intro, Newtonian mechanics. Chap 1: pages 1-3.
9/11: intertial frames, conservation laws. Chap 1: pages 4-6.
9/13: Energy conservation, conservative forces in 1d. Chap 1: pages 6-8.
9/15: 1d motion, functionals and functional derivatives. Chap 1: pages 8-9. Chap 2: pages 1-3.
9/15 afternoon: Euler-Lagrange equations, 1st integral, example. Chap 2: pages 4-6.
9/18: Constraints. Chap 2: pages 6-8.
9/20: Constraints: example. Action and Lagrangian. Chap 2: pages 8-9. Chap 3: pages 1-2.
9/22: Lagrangians: examples. Chap 3: pages 3-5.
9/27: Non-inertial frames. Chap 3: pages 5-6.
10/2: Rigid body example, intro Noether’s theorem. Chap 3: pages 7-9.
10/4: Noether’s theorem. Chap 3: pages 9-11.
10/6: Constraints: Chap 3: pages 12-14.
10/6 afternoon: Falling ladder. Chap 3: pages 14-16.
10/10: Falling ladder cont’d, Lagrangian for a charge particle. Chap 3: pages 16-19.
10/11: Two body problem, effective potential. Chap 4: pages 1-3.
10/13: Formal solution, Kepler problem. Chap 4: pages 4-5.
10/13 afternoon: Binet coordinates, conic section solution. Chap 4: pages 6-7.
10/16: Conic sections and Kepler’s laws. Chap 4: pages 7-9.
10/18: Integral solution and closed orbits, start small oscillations. Chap 4: pages 9-11.
10/20: Midterm review session
10/23: Small oscillations, equilibrium and stability, damping. Chap 5: pages 1-4.
10/25: Forced oscillations, Fourier series, Green’s functions. Chap 5: pages 5-8.
10/27: Green’s functions, coupled oscillations, normal modes. Chap 5: pages 9-13.
10/30: Normal modes, examples. Hamiltonian mechanics intro. Chap 5: pages 14-16. Chap 6: page 1.
11/1: Hamilton’s equations, examples. Chap 6: pages 1-3.
11/3: Charged particles, energy conservation, variational principle revisited. Chap 6: pages 4-6.
11/6: Liouville equation and Poincaré recurrence theorem. Chap 6: pages 6-8.
11/8: Poisson brackets. Chap 6: pages 8-11.
11/10: Canonical transformations. Chap 6: pages 11-13.
11/13: Noether’s theorem revisited, action-angle variables. Chap 6: pages 14-16.
11/15: Integrable systems, generalized thermalization. Chap 6: 16-19.
11/17: Hamilton-Jacobi equation, relation to quantum mechanics. Chap 6: pages 20-22.
11/20: Classical field theory. Chap. 8. pages 1-3.

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