**Mechanics (P421), Fall 2020**

**Instructor:** Romain Vasseur, Assistant Professor

office: Hasbrouck 405A

email: rvasseur[at]umass[dot]edu

office hours: TuTh 5:00-6:00pm (on Zoom)

**Lectures: **TuTh 10:00AM – 11:15AM, remote lectures (on Zoom). Zoom lecture recordings will be made available, but real time attendance is strongly recommended.

**Additional Honors 421 Colloquium: **Fridays 5:30PM – 6:20PM (remote class, on Zoom)

**TA:** Jesse Underland

email: junderland[at]umass[dot]edu

**Syllabus: **link

**Textbook:**

- Classical Dynamics of Particles and Systems (5th Edition) Paperback, Author: Jerry B. Marion by Stephen T. Thornton.
- Suggested reading: Mechanics: Volume 1 (Course of Theoretical Physics Series) 3rd Edition by L. D. Landau and E. M. Lifshitz.

**Website:** https://blogs.umass.edu/rvasseur/teaching/. All the teaching materials (lecture notes, problem sets) will be posted on the course webpage. Problem set solutions will also be made available on Moodle.

**Grading:** The course grade will be based 40% on the homework (with the lowest homework grade being dropped), 20%+20% for the two midterm exams, and 20% on the final exam. Late homework will receive 50% credit.

(In exceptional cases when midterm and final exam scores are much higher than the homework score, the homework scores will be ignored) .

**Problem sets:** All problem sets will be posted online on the course webpage only. I will post the solutions on Moodle. Please email your homework to Jesse Underland, in **PDF** format.

**Discussion sessions:** Monday 10.10-11.00am, Tuesday 4.15-5.05pm, led by Jesse Underland. Those discussion sessions will cover problems associated with the class material, including many problems from exams and midterms from previous years.

**Exams:** The midterms and exams will be held “in class”, remotely. All exams will be open book.

Midterm Exam 1: Thursday 10/08, 10:00AM – 11:15AM

Midterm Exam 2: Thursday 11/05, 10:00AM – 11:15AM

Final Exam: TBA

**Course description:**

Advanced course in undergraduate classical mechanics covering Newtonian dynamics and analytic methods. Topics include conservation laws, oscillatory phenomena including damping and resonance, central force problems and planetary orbits, rigid body mechanics, an introduction to the calculus of variation and the principle of least action, generalized coordinates, with Lagrangian and Hamiltonian dynamics.

**Topics:**

- Chapter 1: Matrices, vectors and vector calculus
- Chapter 2: Newtonian Mechanics
- Chapter 3: Small Oscillations
- Chapter 4: Central Force motion
- Chapter 5: Dynamics of a system of particles
- Chapter 6: Analytical mechanics: Lagrangian
- Functionals and variational calculus
- Action and Euler-Lagrange equations
- Symmetries and conservation laws

- Chapter 7: Coupled oscillations
- Chapter 8: Analytical mechanics: Hamiltonian dynamics

**Lecture Notes:**

- Matrices, vectors and vector calculus
- Newtonian Mechanics
- Small Oscillations
- Central Force Motion. See also this Latex Note for a simple derivation of the trajectories in Kepler’s problem.
- Dynamics of a system of particles
- Lagrangian Mechanics
- Coupled Oscillations
- Hamiltonian dynamics

**Problem Sets: ***Please return your problems by email to Jesse Underland*

- HW 1 (due Friday, Sept 4 )
- HW 2 (due Friday, Sept 11)
- HW 3 (due Friday, Sept 18)
- HW 4 (due Friday, Sept 25)
- HW 5 (due Friday, Oct 2)
- HW 6 (due Friday, Oct 16)
- HW 7 (due Friday, Oct 23)
- HW 8 (due Friday, Oct 30)
- HW 9 (due Monday, Nov 9)
- HW 10 (due Friday, Nov 20)

Solutions available on Moodle, or upon request by email.

**Week 1:** Chapter 1: Matrices, vectors and calculus. Required reading in the textbook: Chapter 1 of Marion, including the examples and their solutions. HW1 due Friday, Sept 4 before 6pm (return to Jesse Underland directly).

Honors colloquium: Nabla operator (problems)

**Week 2: **Chapter 2: Newtonian mechanics (lecture notes pages 1-9). We discussed Newton’s laws and their regime of validity, conservative forces, differential equations and Taylor series, and saw a few simple examples of equation of motion. HW2 due Friday, Sept 11. Required reading in the textbook: Chapter 2 until example 2.9, and example #4 (Time-dependent force) in notes (page 9).

Honors colloquium: Delta function and Fourier transforms

**Week 3: **Chapter 2: Newtonian mechanics (lecture notes pages 10-17). We discussed the motion of a charged particle in a magnetic field, harmonic oscillators, conservation theorems and the solution of the motion of a particle in a one-dimensional potential. HW3 due Friday, Sept. 18. Required reading in the textbook: Chapter 2: 2.5 and 2.6. Chapter 3: 3.1, 3.2, 3.3, 3.4 and 3.5. Also read the solution of the harmonic oscillator using conservation of energy and separation of variables on page 17 of the notes.

Honors colloquium: Fourier transforms

**Week 4: **Chapter 3: Small oscillations. (Chapter 3 in Marion). We discussed stable and unstable equilibrium positions in generic potentials, small oscillations, damped oscillations, as well as forced oscillations. HW4 due Friday, Sept. 25. Required reading in Marion: Chapter 3: 3.6 and 3.8 (including example 3.6), and chapter 8: 8.3, 8.4 and 8.7.

Honors colloquium: Fourier transforms 2

**Week 5: **Chapter 4: Central force motion. We discussed the notion of effective potential, presented the general solution of Kepler’s problem, and conic sections. HW5 due Friday, Oct. 2. Required reading in the textbook for next week: Chapter 9: 9.1, 9.2, 9.3, 9.4, 9.5 and 9.6.

Honors colloquium: Green’s functions

**Week 6: **Chapter 5: Dynamics of a system of particles. We solved the two-body problem, and discussed relative coordinates, along with some aspects of scattering, and of systems of many particles. HW6 due Friday, Oct. 16. Required reading in the textbook for next week: 11.1, 11.2, 11.3 and 11.4.

Honors colloquium: Stability of circular orbits

**Week 7: **Midterm practice on Tuesday, midterm #1 on Thursday. Required reading in the textbook: Chapter 6 (all sections).

Honors colloquium: Precession

**Week 8: **Chapter 6: Lagrangian mechanics. We discussed variational calculus and the concepts of Lagrangian and action. We also went through simple examples of Euler-Lagrange equations of motion. HW7 due Friday, Oct 23. Required reading in the textbook: Chapter 6 (all sections) and Chapter 7: 7.1, 7.2, 7.3 and 7.4.

Honors colloquium: Precession + Soap films

**Week 9: **Chapter 6: Lagrangian mechanics. (cont’d) We discussed how to take into account constraints using Lagrange multipliers. We also started a general discussion of conservation laws, and proved Noether’s theorem. HW 8 is due Oct. 30. Required reading in the textbook: Chapter 7: 7.5 (Lagrange multipliers) and 7.9 (conservation laws); and Chapter 12 (coupled oscillations): 12.2 and 12.4. Next week we’ll give examples illustrating Noether’s theorem, and discuss coupled oscillations.

Honors colloquium: Brachistochrone

** Week 10: **Noether’s theorem, and chapter 7: coupled oscillations. We discussed the general formalism of coupled oscillations using the Lagrangian formalism, as well as simple examples. HW 9 is due Monday, November 9. Next week will be devoted to midterm #2 (practice on Tuesday, exam on Thursday).

Honors colloquium: Wave equations using Lagrangians. Classical field theory.

**Week 11: **Midterm practice on Tuesday, midterm #2 on Thursday.

Honors colloquium: Guitar string and coupled springs.

**Week 12:** We finished the discussion of coupled oscillations, and talked about Hamiltonians, Hamilton’s equations and examples. Required reading: 7.10 and 7.11 in the textbook. HW 10 Due Friday Nov 20.

** Week 13: **Hamiltonian dynamics, Poisson brackets.

Honors colloquium: Classical field theory.