# Class Title: **Mechanics I**

Session: Fall 2018 (09/04/2018 - 12/12/2018)

Days & Times: Tu Th 10:00AM - 11:15AM

Location: Hasbrouck Laboratory Add room 126

Midterm Exam 1: Thursday 10/11, 10:00AM - 11:15AM at HAS Add 126

Midterm Exam 2: Thursday 11/8, 10:00AM - 11:15AM at HAS Add 126

Final Exam: Friday 12/14/2018, Friday, 10:30AM - 12:30PM at HAS Add 126

Additional **Honors 421** Colloquium *(welcome to enroll!)*

Days & Times: Fridays 5:30PM - 6:20PM

Location: Hasbrouck Lab Add room 109

Instructor: Tigran Sedrakyan

E-mail: tsedrakyan@physics.umass.edu

Office: HAS 403B

Phone: 413-545-2409

Office hours: W 5:10-7:15pm at HAS 409. We can meet also after the lectures or other times by appointment.

Discussion Section Instructor: Shuang Zhou

E-mail: zhou@physics.umass.edu

Office: HAS 407A

Phone: 413-545-1291

Days & Times: MW 7:00pm - 8:15pm

Location: LGRT 1033

TA: Guanchu Chen

E-mail: guanchuchen@umass.edu

Office: HAS 113

Office hours: Tuesdays & Thursdays 7:30pm - 9pm

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.

Prerequisites: PHYSICS 151 or 181, MATH 233.

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.

Grading: grades from the various components of the course will determine the final grade. These are weighted as follows:

— Homework solutions -- 40% (with the lowest homework grade being dropped)

— Midterm exam No 1 ( Tuesday 10/9, 10:00AM - 11:15AM at HAS Add 126, during the regular lecture hour) -- 20%.

— Midterm exam No 2 ( Thursday 11/8, 10:00AM - 11:15AM at HAS Add 126, during the regular lecture hour) -- 20%.

— Final exam -- 20%.

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

For Disability Accommodation and Academic Honesty policy statements please see:

*Academic Honesty Policy Statement*

*Disability Statement*

*Where are we in the textbook? -- Required reading -- Homework assignments and solutions -- Notes*

*Where are we in the textbook? -- Required reading -- Homework assignments and solutions -- Notes*

**Week 1
**Matrices, Vectors, and Vector Calculus.

Required reading: Chapter 1; Example problems 1.1 through 1.8 and their solutions. See also Lecture 1 notes and Lecture 2 notes.

Homework assignment 1 (Due Sept 13)

**Week 2
**Newtonian Mechanics. Equations of motion, central forces, and central potentials; types of potentials. A solution of the equation of motion in some special cases. Required reading: Chapter 2; Example problems 2.1 through 2.10 and their solutions. See also Lecture 3 and Lecture 4 notes.

Homework assignment 2 (Due Sept 21)

Honors Colloquium Lecture 2 notes

**Week 3**

The motion of a particle in a magnetic field. Simple harmonic oscillator. Required reading Chapter 2 example problem 2.10, Chapter 3, paragraphs 3.1, 3.2; Example problem 3.1; and Lecture 5 notes. In Lecture 6 we discussed (i) Forced Oscillations: Beats and resonance. (ii) How to use complex variables to solve various equations of motion. (iii) Damped Oscillations.

Homework assignment 3 (Due Sept 28)

**Week 4**

We discussed Damped Oscillations, specifying underdamped, overdamped, and critically damped regimes. Then considered a damped oscillator subject to a periodic force: Forced oscillations with damping (Marion paragraph 3.6), Required reading Chapter 3 paragraphs 3.3 through 3.9 (3.9 is optional). Example problems 3.2 through 3.6. Suggested reading: Chapter 5 paragraphs 21, 22 of Landau and Lifshitz (see the textbook for suggested reading).

In Lecture 8 we discussed Principle of Superposition -- Fourier Series (Marion paragraph 3.8) and Conservative systems (Marion paragraphs 2.5, 2.6). Example problems 2.11, 2.12. Suggested reading: Chapter 2 paragraphs 6 through 9 of Landau Lifshitz. See also Lecture 8 notes.

Homework assignment 4 (Due Oct 5)

Honors section: Analytical techniques for calculation of integrals (implications to Dirac-delta function).

**Week 5**

We discussed the conservation theorems. Required reading: paragraphs 2.5, 2.6.Example problems 2.11 through 2.13. Then we discussed representations of various quantities using polar coordinates. Required reading Chapter one paragraph 1.14, 1.15. Example problem 1.8. See also Lecture 9 notes. Central-force motion. Required reading: Chapter 8 paragraphs 8.1 through 8.7. Suggested reading: Chapter 3 paragraphs 14,15 of Landau Lifshitz.

Central-force motion. Planetary motion - Kepler's problem. Required reading: Chapter 8 paragraphs 8.1 through 8.7. Suggested reading: Chapter 3 paragraphs 14,15 of Landau Lifshitz. See also Lecture notes10.

Honors section: Honors Lecture notes

No Homework assignment this week.

Practice Midterm1a: problems; solutions.

Practice Midterm1b: problems; solutions.

**Week 6**

No lecture on Tue, Oct 9 (because of the Columbus day holiday).

Midterm on Thursday, Oct 11.

Honors section: the perturbative approach to nonlinear oscillations.

Homework assignment 5 (Due Oct 19)

**Week 7**

Closed orbits, small perturbations around circular orbits. Marion paragraphs 8.8, 8.9, 8.10. Example problems 8.5 through 8.7. See also See also Lecture notes 11. Dynamics of a system of many particles: Center of mass, linear momentum and the energy of the system, two-body problem, the scattering of two particles, totally inelastic scattering, elastic collision. Required reading: Chapter 9, paragraphs 9.1, through 9.8. Example problems 9.1 through 9.10.

Honors section: Scaling and Mechanical Similarity. Required reading: Landau-Lifshitz, Chapter 2 paragraph 10

Homework assignment 6 (Due Oct 30)

**Week 8**

Systems of many particles. Require reading: Chapter 9, paragraphs 9.1, through 9.8. Example problems 9.1 through 9.10. Dynamics of rigid bodies: the moment of inertia, angular momentum. Required reading: Chapter 11, paragraphs 11.1 through 11.4, example problems 11.1, 11.2, 11.4, 11.4. Lagrangian of a single particle in an external central potential field. Marion 6.1, 6.2, 7.1, 7.2. Landau, Lifshitz 1 & 2. Derivation of Newton's equations of motion from Lagrangian mechanics. Required reading Required reading: Chapter 7, Chapters 7.1 through 7.4; 7.6.

Honors section: Theorems in Lagrangian mechanics. Derivation of the Lagrangian for a free particle using Galileo's relativity principle.

Homework assignment 7 (Due Nov 2)

**Week 9**

Curves in higher dimensions, Functionals. Calculus of variations: Euler-Lagrange equation, Lagrangian mechanics. Required reading Chapter 6, paragraphs 6.1 through 6.4, example problems 6.1, 6.2, 6.3. Lagrangian mechanics/Dynamics, Generalized coordinates, Euler-Lagrange equations of motion in generalized coordinates, the principle of the least action. Required reading Chapter 6, paragraphs 6.3, 6.4, example problems 6.1, 6.2, 6.3.

Honors Section: Alternative form of Euler's equation. Calculus of variations: shape and the potential energy of a flexible cable suspended from two fixed points.

No Homework assignment this week.

**Week 10**

Small (coupled) oscillations. Two coupled harmonic oscillators, weak coupling, the general problem of coupled oscillations. Required reading Chapter 12, paragraphs 12.1 through 12.4. Example problem 12.1. See also Lecture notes.

Practice midterm 2a: Problems and solutions

Practice midterm 2b: Problems and solutions

Honors Section: Problems in Lagrangian Mechanics.

Homework assignment 8 (Due Nov 16)

**Week 11**

Small oscillations in coupled systems with many degrees of freedom, example problems. Conservation theorems revisited: conservation of energy, conservation of linear momentum, conservation of angular momentum. Required reading: Chapter 7, paragraph 7.9. Euler-Lagrange equations with multipliers: forces of constraints. Required reading: Paragraph 7.5. Example problems 7.9, 7.10. See also Lecture notes.

Honors Section: the Lagrangian approach to relativistic mechanics.

**Week 12**

Thanksgiving recess

Homework assignment 9 (Due Nov 30)

**Week 13**

The Hamiltonian: Canonical equations of motion - Hamiltonian dynamics. Required reading: Chapter 7, paragraphs 7.10 through 7.11. Example problems: 7.11, 7.12. Energy conservation and conservation theorems in the Hamiltonian formulation; The phase space (Paragraph 7.12). See Lecture notes (pages 1-6) Poisson brackets and their properties. See Lecture notes (pages 7-10). Fundamental Poisson brackets and relation to canonical quantization. See notes.

Honors Section: Problem-solving session in Hamiltonian mechanics.

Homework assignment 10 (Due Dec 11)

**Week 14**

Perturbation theory. The perturbative solution of nonlinear (anharmonic) equations of motion. Phase diagrams of nonlinear systems. Self-limiting equations/systems. Chaos. See notes.

Honors section: Problems in Hamiltonian mechanics.

**Week 15**