Mechanics (P421), Fall 2019

Mechanics (P421), Fall 2019

Instructor: Romain Vasseur, Assistant Professor 
office: Hasbrouck 405A
email: rvasseur[at]umass[dot]edu  
office hours: TuTh 4:30-5:30pm. Or by appointment and email, or visit my office

Lectures: Tu Th 10:00AM – 11:15AM, room: Hasbrouck Laboratory Add room 126

Additional Honors 421 Colloquium:  Fridays 5:30PM – 6:20PM
Location: Hasbrouck Lab Add room 106

TA: August Miller
email: augustmiller[at]umass[dot]edu
Office: LGRT 1129, Mailbox: LGRT 1127A
Office hours:   Mo 4:30-6:00pm

Syllabus: syllabus_p421f19


  • 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: Problem set solutions will also be made available on Moodle. 

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

Problem sets: The problem sets will be “paperless”, and posted online on my website only. I will post the solutions on Moodle.

Midterm Exam 1: Thursday 10/24, 10:00AM – 11:15AM in HAS Add 126
Midterm Exam 2: Tuesday 11/19, 10:00AM – 11:15AM in HAS Add 126
Final Exam: Thursday 12/19, 10:30AM – 12:30PM in HAS Add 126

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.


  • 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:

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

Problem Sets: Please return your problems to August Miller’s mailbox on the 11th floor of LGRT in room 1127A. (Go right when you get off the elevator, then take the first left. His mailbox is roughly in the middle and is labeled “MILLER.”)

  1. HW 1 (due Friday, Sept 13 )
  2. HW 2 (due Friday, Sept 20 )
  3. HW 3 (due Friday, Sept 27)
  4. HW 4 (due Friday, Oct 4)
  5. HW 5 (due Friday, Oct 11. Update: Now due Tuesday, Oct 15 by 5pm.)
  6. HW 6 (due Monday, Oct 21 by 5pm)
  7. HW 7 (due Friday, Nov 8)
  8. HW 8 (due Friday, Nov 15)
  9. HW 9 (due Tuesday, Dec 3)
  10. HW 10 (due Tuesday, Dec 10)

Solutions available on Moodle, or upon request by email.

Midterm and final Exams: 

  • Practice Midterm #1 (Thursday Oct 17 4–6pm, MOR1N326) from 2016 and 2017
  • Review session #1 (organized by August Miller, Tuesday Oct 15, 4.30 to 6.30pm in LGRT 1033): Problems
  • Midterm 1 (Midterm1)
  • Practice Midterm #2 (Thursday Nov 14, 4:30pm – 6:00pm, Has 113) from 2016 and 2017
  • Review session #2 (organized by August Miller, Tuesday Nov 12, 4.30 to 6.00pm in Has 242): Problems
  • Midterm 2 (Midterm2)
  • Practice final exam from 2018

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 13 before 6pm (return to August Miller 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 20 before 6pm (return to August Miller directly).  Required reading in the textbook: Chapter 2 until example 2.9.

Honors colloquium: Delta function and Fourier transforms

Week 3: Chapter 2: Newtonian mechanics (lecture notes pages 10-18). 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. 27 before 5pm. 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.

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, forced oscillations, as well as the superposition principle and Fourier series. HW4 due Friday, Oct. 4 before 5pm. 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: Chaos, logistic map

Week 5: Chapter 4: Central force motion. We discussed the notion of effective potential, presented the general solution of Kepler’s problem, and discussed Kepler’s laws. HW5 due Friday, Oct. 11 before 5pm. 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 systems of many particles.  HW6 due Monday, Oct. 21 before 5pm. 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: No class on Tuesday. Discussion session organized by August Tuesday Oct 15, 4.30 to 6.30pm in LGRT 1033. Midterm practice on Thursday Oct 17 4–6pm, MOR1N326. On Thursday, we discussed some aspects of the motion of rigid bodies with an emphasis on moments of inertia.

Honors colloquium: Precession

Week 8: Tuesday: Variational calculus. Thursday: Midterm #1.

Week 9: 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, and started discussing motion in non-inertial frames. HW7 due Friday, Nov. 8 before 5pm. HW 8 also posted, due Nov. 15. Required reading in the textbook: Chapter 6 (all sections) and Chapter 7: 7.1, 7.2, 7.3 and 7.4. Next week we will discuss constraints (section 7.5) and conservation theorems (section 7.9).

Honors colloquium: Soap films

Week 10: Chapter 6: Lagrangian mechanics. (cont’d) We discussed inertial frames (Coriolis and centrifugal forces) using the Lagrangian formalism, and how to take into account constraints using Lagrange multipliers. We also started a general discussion of conservation laws. Recall that HW 8 is due Nov. 15. (no late HW will be accepted). 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 show Noether’s theorem, and discuss coupled oscillations.

Honors colloquium: Guitar string in Lagrangian formalism

Week 11: Discussion session organized by August Tuesday Nov 12, 4.30 to 6.00pm in Has 242. Midterm practice on Thursday Nov 14, 4:30pm – 6:00pm, Has 113. This week we finished Chapter 6 and proved Noether’s theorem. We started discussing coupled oscillations and normal modes (Chapter 7).

Honors colloquium: Geodesics

Week 12: Tuesday second midterm. On Thursday we finished the chapter on coupled oscillations by discussing the normal modes of coupled pendulums. We also started our last chapter on Hamiltonian dynamics, and derived Hamilton’s equations. HW 9 due after the break, on Tuesday December 3. Required reading for after the break: 7.10 and 7.11 in the textbook.

Honors colloquium: Coupled oscillations, waves and classical field theory

Week 13: Thanksgiving break

Week 14: We discussed examples of Hamiltonian dynamics. We also covered the notions of phase space, Poisson brackets and canonical transformations. HW10 is due Tuesday December 10.

Honors colloquium: Action-angle variables

Week 15: Tuesday: final exam review session.

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