PHYSICS 303: Classical Mechanics

Text:

Classical Mechanics , John R. Taylor;

I am happy to talk with you about physics, math, or how they relate to the world, your text, and/or your assigned homework!
The class homepage is at     http://panda.unm.edu/courses/finley/p303.html
The class's Teaching Assistant is not yet determined. He will be available for discussions and/or questions, and will have an office in the building, with a regular office hour once a week at a time yet to be determined.

We study Classical Mechanics for several reasons:

• It is a subject that once was thought very well understood; in fact it was almost all of the content of careful and rigorous physics until sometime in the early 1800's. This was true even though the subject certainly was only well understood when the additional assumption that everything was linear was put into the entire formulation. As well it definitely served as the foundation for all of the rest of physics, discovered since that time.
  However, in the last 30 years, or so, it has been discovered that one actually can solve nonlinear problems in several somewhat universal ways, and that many important regularities can be understood as properties of the world in which we live. In addition a detailed science of chaos, or chaotic motions, has been invented and applied, built also on the foundation of (linear) classical mechanics.
• This subject is also the conceptual and philosophical foundation for all of physics. The ideas of force, momentum, energy, and angular momentum are the foundational notions for relativity, electromagnetism, quantum mechanics, and quantum field theory. As well, the new concepts which you will learn in this course, of Lagrangian and Hamiltonian mechanics, are the basis for all forms of quantum physics. Of course it also provides the foundation for wave motion, which one needs both in electromagnetism and in quantum physics.
• Study of classical mechanics will ALSO provide you a powerful reason to understand why you have needed to take so many mathematics courses, most of which are prerequisites for our study here. The author of our text has taken considerable time to apply many different mathematical ideas to easily-visualized physics problems, to help you acquire motivation to practice (often) those mathematical ideas that you have so far (probably) not had many applications for.
  For many of you this may be your first junior-level physics; therefore, I look forward to working with you and getting to know you better. Your input, in terms of questions and comments, and your work with problems and exams, will help determine our pace, although we will be guided by the weekly syllabus.

The class has a syllabus which should also be taken as a Reading Assignment Guide.
Please use it to read carefully through the material to be covered, BEFORE the lecture, keeping a list of questions that your reading generated, to be discussed in class. Questions are encouraged at all times during any lecture!
Then, re-read it, afterward; it should be easier this second, or (perhaps) third, time.
Doing well in the class requires that you spend at least 10 hours per week on the material for this class, in addition to the time spent in class!

Comments on the text and its author:
The author is very careful to explain the new concepts, often doing it in two or three different ways, to make sure that you will become comfortable with each new thing that he introduces. He takes the same approach with concepts that he feels you may not yet understand particularly well. Lastly, he worries considerably about possible misconceptions that you might have acquired, and tries to talk you out of them. All these explanations are really very good; do try to read through them until you feel he would be happy with your understanding. Also please allow me to help you understand them.
This author believes that the use of coordinate systems other than the usual Cartesian system, using {x,y,z}, makes vectors too complicated---since the unit vectors are no longer constant. I am not quite consistent with this philosophy, which is quite different from what normally occurs in texts on this subject material. Therefore, while we will pursue some middle ground in this regard, spending some time on both cylindrical and spherical coordinate systems, and bases sets for vectors as well.
The author also emphasizes the use of the Lagrangian approach to mechanics, which uses scalar quantities and so-called "generalized coordinates," instead of the (completely equivalent) Newtonian approach to mechanics, which uses vector quantities. We will therefore use this attitude to inform our "middle road," and see how well it will work for us.

Now let me make a few comments about the overall content and intent of the various chapters that we will discuss this semester:

• Chapter 1 is simply a review of Newton's Laws and their relation to vector mechanics. We will go quickly there, reminding you of the relevant material from your earlier course on mechanics, and emphasizing the relationship between those laws and important conservation principles.
  In particular we will review the use of Free-Body Diagrams in fairly sophisticated ways.
  This chapter also gives the simplest possible example of the coordinate form of the important Second Law in a different coordinate system: 2-dimensional cylindrical coordinates, where we will use both those coordinates and their associated basis vectors.
• Chapter 2 re-visits the motion of projectiles under earth's gravity, but this time does NOT ignore the air resistance, so that the results can be more realistic.
  It also looks at the usual circular (or helical) motion of charged particles in a magnetic field. However, it does this from first principles, i.e., from "scratch," using the Second Law. The chapter also takes these problems as opportunities to introduce two new mathematical notions:

  using computers to create better representations of motions in complicated situations,

  using complex numbers, especially inside exponentials, to simplify some ubiquitous algebra.

  We will definitely follow along with his two new notions, and try to help you learn these new tools.
• Chapter 3 is a review of the conservation principles for momentum and angular momentum, and some methods for many-particle systems. In the process of studying systems with more than one (or two) particles, we will discuss total mass, center-of-mass, total momentum, etc.
• Chapter 4 concentrates on the tremendously-important concept of conservation of energy, and its relation to work done, especially that done by non-conservative forces.
  It therefore describes conservative forces, and their related potential energies---which are a property of two interacting bodies.
  It also describes methods to acquire a qualitative understanding of the motions of a body which has some given potential energy, even when the differential equations are not solvable exactly.
  Lastly, it begins a more serious consideration of curvilinear coordinate systems, especially the very common spherical coordinates, which will prove useful for us.
• Chapter 5 describes oscillations of simple systems in considerable detail, which underlies much of more advanced ideas in both classical and quantum mechanics.

  It also introduces the applications of Fourier Series.

• Chapter 7 shows the equivalence of Lagrange's approach to mechanics, which emphasizes the uses of scalars (and generalized coordinates) to solve mechanics problems, instead of the vector forces which are needed in Newton's method.
  The proof of the equivalence of the two methods uses the Calculus of Variations, which is explained in Chapter 6 , so we will look, briefly, at the material there.
  Lastly, we will introduce the notions of Hamilton's approach to mechanics, from Chapter 13. It also emphasizes the use of scalars and generalized coordinates, but concentrating on position and momentum instead of the position and velocity variables used by Lagrange.
• Chapter 8 studies in some detail the two-body, central-force problem, for various different such forces. A very important force is of course the standard, Newtonian gravitational force. We will go into some depth, looking at motions under the influence of this force, and will introduce the idea of the eccentric anomaly as a very useful way to look at their time dependence.
  The chapter also looks at some interesting, real-life perturbations that can affect these systems. We will follow those and probably also spend time on a Jupiter fly-by problem.
• Chapter 9 is concerned with the changes to all our methods that are generated if we decide to operate from a non-inertial reference frame. It considers two basic kinds: linearly-accelerated frames, and rotating frames. This second kind involve centrifugal and Coriolis forces, so we can find some earth-bound applications.

Help with the Course

The key to passing this class is to understand the basic concepts well enough that you can solve the homework problems, and know why you made the choices and decisions that you did in solving the problems! Please, after you have solved each problem, go back and think about why it worked. If you cannot decide that, ask me questions about it, look at the web-posted solution, or both. When you can do this---solve the homework problems, and understand which concepts were involved in doing this---you should have little trouble with the exams.
Be sure and work through ALL the examples in the textbook. That means putting your pencil to the paper; not just reading through the words in the book, and saying you "understand" them. The examples are very good at trying to explain what the "Key Ideas" are. Try to "guess," in advance what those key ideas will be, and then check to see if you were correct.
Note that there are very many homework problems at the end of each chapter! I will certainly not assign all of them, nor will we work through all of them in the problem sessions; therefore, pick out some additional ones to work through yourself---in your study group---from each section of each chapter. I will happily help you with them if your group becomes stuck.
Lastly, pay very good attention to the Summaries of Key Ideas that the author has provided at the end of each Chapter.

The exams will have questions that involve the same concepts as did the homework problems; many times they will look a lot like the homework problems, or the examples in the text, or problems I have worked in class, or---and perhaps even especially---the problems we have solved together in the weekly problem sessions. Sometimes the questions will ask that you explain the concepts in words, or choose between alternative interpretations of the concepts.

Although I'd like to think I can explain things well enough that you'll understand everything the first time in the lecture---having already read the material before that lecture--- that sometimes does NOT happen.

The only way to really learn the physics is by sweating through the problems and examples, making lots of mistakes, getting hopelessly stuck at times, and even experiencing profound frustration. (Talk with someone else at that point.) This method is a hard route, but it goes with the territory, of wanting to become a physicist. You should allocate plenty of time to read the text (twice!) and to do the assignments. Please do them at a time when you are alert with no distractions. Setting aside one hour before the football game to do your physics homework is not a good idea. Even worse is trying to do them 1 hour before they're due. If you can't devote the necessary time, you really shouldn't be taking this course. I'd like to see everybody succeed, but that's only possible if you can make the commitment and put in the time. The reward for all this effort, hard work, and frustration on your part is that you will have developed the problem solving skills, work ethic, and discipline that will be very important for the rest of your life---and, of course, you will have passed this course!

• The weekly Problem Session, Phys. 451-054, for this course meets on Tuesday from 7 to 9 pm, in the same room as our usual class. The purpose is to help you learn the material from the more "regular" portion of the class, i.e., classical mechanics.
  Format for this time will vary, although the "standard" pattern will be that I will always first answer questions that you might have. Then you will work problems that I have chosen, in small groups, on the blackboard; I will select those problems based on the material under discussion in recent lectures. The TA and myself will circulate among these groups, helping with details as needed. We will then have presentations and questions concerning how the solutions were obtained.
  In case you decide you want to spend some additional time studying those problems discussed, there is a webpage, accessed from this link, that lists the problems, after they have been worked out during the session.
  A few times, including the first one of the semester, I will use that time to present some review of mathematical details that we will be needing in the class.
  As well, four of these times will be used to take the 4 examinations for the (regular) class.
• Outside of class time you may also ask questions of me, or the TA. In addition to questions about physics, and mathematics, I can usually answer questions about the use of Maple on computers.
• It is quite important that you begin to learn to work together in groups, with a questioning mode of each of the participants. This is the way that physics problems are solved in large scientific installations, and is a very valuable way to learn: each of you will often bring a different insight to the working out of the problem. The groups created in a temporary way for the weekly problem sessions might help this process.
• There is an undergraduate lounge in the building---at the far west end---which has sofas, tables, computers, and printers.

• Homework:
  As noted on the main class webpage, there will be homework due on (almost all) Fridays and Mondays (except for those Mondays that are just prior to an exam on Tuesday). It is to be turned in, on the front desk, as you come into class on the day that it is due. We will often discuss one or more of the problems during class that day; therefore, it is important that you turn it in as you enter the classroom, even if you have come late. Note that my accepting late homework is a very unusual occurrence!
  The various assignments may be found on the appropriate webpage at least by the time the previous one is due, hopefully sooner than that. Computer calculations made for homework should be accompanied by a printout of the output of the program. Graphs requested for homework should be computer printed.
  Homework will be graded by the class Teaching Assistant, with each problem as either 5 or 10 points, depending on the length and difficulty. Questions as to the grading should FIRST be directed to the Assistant, either during his office hour or after the weekly problem session.
  A Bonus Problem will be assigned about once per month. They are somewhat challenging, but often more interesting. They will have a pre-assigned number of points attached, and will be graded somewhat more rigorously than the regular homework, by myself. The grades earned there will be added on top of the final homework score, allowing you to have a final average score higher than 100%.
• Exams:
  As noted elsewhere, there will be 4 examinations during the course of the semester. The questions on them will be similar, but sometimes slightly "beyond" the homework problems, or the problems worked through in the problem sessions, or the examples in the text, or the additional examples that I work through during class time. ["Beyond" means that they will use the knowledge obtained during the working out of those problems, but request that you apply that knowledge to new situations.]
  Also as already noted, they will be concerned with finding out if you understand "why" you should apply various formulas for various situations. I am NOT concerned that you memorize all these formulae; therefore, the exams will allow you to bring with you one page of (the usual size of) paper with anything on it that you have yourself written there, by hand. The exams are NOT like "high school." There is no intent that the grades should be 90 to 100 is an A, etc. Instead the intent is to give you clues as to how well you are integrating the concepts of physics into problem solving in physics. Therefore, the letter grades on exams are "curved" to match with what you are telling me about your understanding. The more actual, correct "work" that you can put on an exam question, relevant to that question, the more partial credit I will assign.
  The examinations will be given during the regularly-scheduled time for the problem sessions, as noted in the Syllabus. They will take up at least almost all of the regular 110 minutes that the sessions are scheduled to last.
  The lowest of the 4 exam grades will be dropped before final grades are computed. As a result of this, there will NOT be any makeup exams.
• The Final Exam:
  The final examination is required of all students. Roughly half of it will concern the material from Ch. 9, which has not previously been covered on exams, while the other half will be a cumulative review of the earlier portion of the semester.
  It is scheduled by the university to occur on 16 December, Tuesday, 7:30-9:30 am (i.e., in the early morning).
• Computation of Grades:
  At the end of the semester, each student will have created 6 grades
  1. one for the average of all scores for the Homework, including the optional Bonus Homework Problems that you chose to turn in.
     
  2. grades for each of the four exams,
  3. and the score for the Final Exam, which is required.

  To determine your complete grade, I will drop the lowest of each student's four Exam grades. The remaining 5 grades will then each count 20% toward your complete grade.
  Each of these 5 grades will be "curved," so that the class average becomes a grade somewhere near C+.
  The complete grade made from the 5 grades described above will also be curved, with the class average for it treated as above. Therefore, the individually-curved grades should be considered as "good" approximations, only, relative to the complete grade determined at the end.

   If you are a qualified person with disabilities who might need appropriate academic adjustments, please communicate with me as soon as possible so that we may make appropriate arrangements to meet your needs in a timely manner. Frequently, we will need to coordinate accommodating activities with other offices on campus.