## Classical Mechanics (P601), Fall 2022

Classical Mechanics (P601), Fall 2022

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.

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

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:

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

Problem sets:

1. HW1 (due Sept 21)
2. HW2 (due Sept 30)
3. HW3 (due Oct 7)
4. HW4 (due Oct 14)
5. HW5 (due Oct 28)
6. HW6 (Due Nov 4)
7. HW7 (Due Nov 18)
8. HW8 (Due Dec 2)

Schedule:

9/7: Introduction. Newtonian mechanics. Chap 1: pages 1-3.
9/9: Conservation theorems. Chap 1: pages 4-6
9/9 afternoon: Conservative forces in 1d. Chap 1: pages 7-9.
9/19: Functionals and Euler-Lagrange equation. Chap 2: pages 1-4.1
9/21: First integral of Euler-Lagrange equation, constraints. Chap 2: pages 5-7.
9/23: Constraints. Actions and Lagrangians. Chap 2: pages 7-9. Chap 3: pages 1.
9/23 afternoon: systems of many particles, center of mass. Lagrangian mechanics: general approach, example: central forces. Chap 1: pages 9-11. Chap 3: pages 2-3.
9/26: Examples. Non-inertial frames in Lagrangian mechanics. Chap 3: pages 4-5.
9/28: Non-inertial frames, rotation of rigid bodies. Chap 3: pages 5-7.
9/30: Noether’s theorem: proof, energy and momentum. Chap 3: pages 8-10.
10/3: Rotation invariance and angular momentum. Constraints, Atwood machine. Chap 3: pages 10-13.
10/5: Constraints: Falling ladder example. Chap 3: pages 13-15.
10/7: Falling ladder (cont’d), Lagrangian of a charged particle. Chap 3: pages 16-18.
10/14: Lagrangian of a charged particle. Chap 3: pages 18-19. Two body problem: Chap 4: pages 1-2.
10/17: Effective potential, Kepler’s problem and integral solution. Chap 4: pages 3-5.
10/19: Binet equation, conic sections. Chap 4: pages 6-8.
10/21: Kepler’s laws, integral solution revisited. Chap 4: pages 9-11.
10/24: End Kepler, Small oscillations, damped oscillations. Chap 5: pages 1-3.
10/26: Forced oscillations. Chap 5: pages 3-5.
10/28: Resonances, Fourier series, Green’s functions. Chap 5: pages 6-8.
10/31: Green’s functions, coupled oscillations. Chap 5: pages 9-11.
11/4: Coupled oscillations, normal modes, example. Chap 5: pages 11-14.
11/7: Coupled pendulums, Hamiltonian. Chap 5: pages 15-16. Chap 6: page 1.
11/9: Hamilton’s equations, examples, charged particle. Chap 6: pages 1-4.
11/14: Charged particle, energy conservation, least action principle revisited. Chap 6: pages 4-6.
11/16: Liouville theorem, Liouville equation, Poincare’s recurrence theorem. Chap 6: pages 6-7.
11/18: Poincare theorem cont’d. Poisson Brackets. Chap 6: pages 8-10.
11/18 afternoon: Classical field theory. Lagrangian and Hamiltonian densities.
11/21: Canonical transformations. Chap 6: pages 11-13.
11/22: Infinitesimal canonical transformations, generators and Noether’s theorem revisited. Action-angle variables: example. Chap 6: pages 13-15.
11/28: Action-angle variables, integrable systems, examples, Thermalization. Chap 6: 16-18.
11/30: Generalized Gibbs ensembles, Hamilton-Jacobi equation. Chap 6: pages 18, 20-21.
12/2: Relation to quantum mechanics, sum over paths and action. Rigid bodies: Kinematics. Chap 6: page 22. Chap 7: pages 1-2.