Advanced Mechanics (PHYS*3400)

Course Outline

Objectives of this course

This course covers some advanced techniques of classical mechanics, focusing mostly on the Lagrangian and Hamiltonian formulations of the laws of mechanics. It begins with a review of the Newtonian formulation, and ends with a couple of lectures on frontier aspects. It will use the mathematical language MATLAB or its free equivalent version FREEMAT to support the numerical solution of some problems. The formal prerequisites for this course are PHYS*2450 (Mechanics II) and either one of MATH*2170 (Differential Equations I) or MATH*2270 (Applied Differential Equations).

Instruction

Lectures

Mid-term Examination: to be scheduled after the reading week
Final Examination: Sat, April 16th, 7pm-9pm

Course Materials

Text

”Advanced Mechanics” (2008), lecture notes by Eric Poisson. http://www.physics.uoguelph.ca/poisson/research/mech.pdf or printout available in the Quiz room ( NSC 1101A)

Course web page

There will be a course web page at http://www.physics.uoguelph.ca/~ralf/PHYS3400.
The main source of information will be the course web page to be set up at D2L providing you with the problem sets, notes about MATLAB and access to your grades.

Course content

1. Newtonian mechanics
   1- Reference frames. 2- Alternative coordinate systems. 3- Mechanics of a single body. 4-Mechanics of a system of bodies. 5- Kepler's problem. 6- Numerical integration of differential equations.
2. Lagrangian mechanics
   1- Introduction: From Newton to Lagrange. 2- Calculus of variations. 3- Hamilton's principle of least action. 4- Applications of Lagrangian mechanics. 5- Generalized momenta and conservation statements. 6- Charged particle in an electromagnetic field. 7- Motion in a rotating reference frame.
3. Hamiltonian mechanics
   1- From Lagrange to Hamilton. 2- Applications of Hamiltonian mechanics. 3- Liouville's theorem. 4- Canonical transformations. 5- Hamilton-Jacobi equation.
4. Frontiers
   The last week of lectures, time permitting, will be devoted to a selection among the following topics: chaotic dynamics, the transition from classical mechanics to quantum mechanics, the principle of least action for fields, the principle of least action for relativistic particles and (super)strings, etc.

References

You may find it useful to consult the following books:

• Herbert Goldstein, Charles P. Poole, and John L. Safko, Classical Mechanics (3rd Edition) (Addison Wesley, 2002; ISBN 0201657023; QA 805.G6)
• E. M. Lifshitz and L. D. Landau, Course of Theoretical Physics: Mechanics (3rd Edition) (Butterworth-Heinemann, 1982; ISBN 0750628960; QA 805.L283)
• Stephen T. Thornton and Jerry B. Marion, Classical Dynamics of Particles and Systems (5th Edition) (Brooks Cole, 2003; ISBN 0534408966; QA 845.M38)
• Joseph L. McCauley, Classical Mechanics: Transformations, Flows, Integrable and Chaotic Dynamics (Cambridge University Press, 1997; ISBN 0521578825; QC 125.2.M39)
• Gregory L. Baker and Jerry P. Gollub, Chaotic Dynamics: An Introduction (2nd Edition) (Cambridge University Press, 1996; ISBN 0521476852; QA 862.P4B35)
• Cornelius Lanczos, The Variational Principles of Mechanics (4th Edition) (Dover Publications, 1986; ISBN 0486650677; QA 845.L3)

Evaluation

Grades will be based on homework assignments, a term project, a laboratory experiment, a closedbook midterm exam, and a closed-book final exam. You will be provided with a one page formula sheet for the exams that you can use also for additional handwritten notes. The scheduling of the midterm exam will be decided during the first few weeks of class.

The final mark will be the best of the two alternate marking schemes for midterms and finals. No other marking scheme will be considered.

I expect to distribute about three or four sets of homework problems to be worked out on a time scale of about two weeks. You are permitted to discuss the homework problems with your colleagues while trying to solve them. However, and this is important, after the discussions you must write up the solutions yourself, independently of anyone else. Cheating will not be tolerated. After each assignment has been handed in, the solutions will be posted on these web pages.

The term project is essentially a long assignment that is attached at the end of the lecture notes. It must be completed before the last week of classes. The term project includes a computational component in which you will be asked to solve a mechanical problem using numerical methods.

The laboratory experiment is a surprise!

Notes about the content by Eric Poisson

You will master other areas of mathematics as well, such as Taylor expansions, total and partial differentiation, line integration, complex algebra, trigonometric functions, etc.

This is a demanding course in which you will learn sophisticated methods of theoretical physics. These are useful well beyond the context of this course; Lagrangian methods apply, for example, to field theories such as electromagnetism, the standard model of particle physics, and general relativity. This is physics for grown-ups, and it is imperative that you conduct yourself as a grown-up while taking this course. This means, most of all, taking responsibility for your own
education. In practice, this means that you will spend a considerable amount of time outside of class hours teaching yourself the material.

The mastery of physics does not come easily to anyone. You must work hard at it, and the work must be done by you, and no one else. My role as your instructor is to help you in this process. I do not believe, however, that I can truly teach you the material. I can lecture on it, give you an orientation, and illustrate the basic ideas, but all this will not make you learn the material. (This may only give you a false impression that you have learned it.) What you must do instead is teach yourself, and my role as your instructor is mostly to help you teach yourself. Knowledge and understanding will come from your own efforts; they cannot be directly transmitted from me to you. It is imperative that you come prepared to do your own learning in this course, and be willing to spend the time that this requires.

Out of necessity, my lectures will be given at a level that is lower than the coverage in the lecture notes. My purpose in the lectures is to introduce the main themes and talk about them in fairly general terms. But attending the lectures will not be enough; you will be expected to master the material at the higher level set in the lecture notes. It will be your job to go through the lecture notes in detail, outside of class hours, following the schedule that I will give in class. You are
expected to read the text carefully, go through all the mathematical steps, fill the gaps, and work through the exercises that are scattered throughout the notes. Doing all this will prepare you very well to do the problems at the end of each chapter, and solving these problems will prepare you very well for the midterm and final exams.

If you follow this course of action you will do very well in this course. And if you follow this advice in other courses as well, your education in physics will be rock-solid. If, however, you fail to read the notes, go though the math, do the exercises and problems, or if you rely too much on group work, then you will acquire no useful skills from this course. And if this is your general attitude, your education in physics will turn out to be useless. Physics is challenging. Live up to the challenge!

Course Policies

The Department of Physics requires student assessment of all courses taught by the Department. These assessments provide essential feedback to faculty on their teaching by identifying both strengths and possible areas of improvement. In addition, annual student assessment of teaching provides part of the information used by the Department Tenure and Promotion Committee in evaluating the faculty member's contribution in the area of teaching.

The Department's teaching evaluation questionnaire invites student response both through numerically quantifiable data, and written student comments. In conformity with University of Guelph Faculty Policy, the Department Tenure and Promotions Committee only considers comments signed by students (choosing "I agree" in question 14). Your instructor will see all signed and unsigned comments after final grades are submitted. Written student comments may also be used in support of a nomination for internal and external teaching awards.

NOTE: No information will be passed on to the instructor until after the final grades have been submitted.