Instructor

Professor William Loinaz
Office: 223 Merrill Science Center
Phone: (413) 542-7968
email: waloinaz [AT] amherst [DOT] edu
Office Hours: TBA
You're also welcome to drop by my office. If I'm in and free I'm happy to meet with you, but it's not so often during the day that I'm in and I'm free.

Grader: Matthew Mo

Course Information

Course Catalog Description:

Schedule

• Class: Tuesday and Thursday 10:00-11:20, Merrill 211
• Problem Session (informal): Wednesdays and Thursdays at 10 pm, Physics student lounge
  

Requisites

Course requirements

• Attendance: Come to class. Lots of stuff will happen there and you wouldn't want to miss it. Lectures and in-class discussion will generally NOT duplicate the text, so don't assume you can just "get it from the book". It's your affair how you allocate your time and I won't explicitly penalize you for being absent, but if you're going to be absent please do me the courtesy of letting me know in advance, if possible. It's a small class--I'll notice if you're missing. And if you're consistently absent I'll want to know why. If you arrive late, please enter quietly at the back of the classroom. Obviously it will be your responsibility to find out what happened in the class you missed.

• Homework: I'll assign weekly problem sets, due Thursdays @11:59 pm. You may leave your homework in my mailbox near the physics department office, or of course you may give it to me personally (please do NOT leave it in the box beside my office or stick it under my door). Homework submitted late without prior arrangement will receive a 50% penalty if submitted within three days of the due date (i.e. by Sunday @11:59 pm), and will receive no credit after that.

  Why do the homework?
  Since you're sophomores and juniors, you've all figured this out by now. But, I'll say it anyway...

  I can't emphasize enough the importance of working the problems. In some of your classes homework is primarily evaluative; the point is for you to demonstrate what you've learned from the readings and lectures. In physics the homeworks are primarily instructional; you learn physics primarily by doing working problems. You must work the problems, think about the results, and understand any mistakes you've made if you wish to attain the type of understanding of the subject required of a working physicist. In at nutshell: If you can't work problems you don't know physics. I (or a grader) will grade the problems, and I'll hand out solutions. I encourage you to read the solutions and understand any mistakes immediately. If something doesn't make sense, ask me about it right away---don't wait until right before an exam.

  Homework extensions
  If you've got a good reason why you need an extension, come talk to me in advance. I'll usually grant the extension for some additional reasonable amount of time that we agree upon. However, I will not grant a homework extension without penalty if you ask for it on the day the homework is due, so don't ask for one. In general, life will be easier for both of us if you do your best to finish the problem set on time and hand in as much as you've been able to complete by the deadline. [If you need such a last-minute or post-facto extension due to extenuating circumstances (e.g. death in the family, sudden illness, travel problem), consult the Dean of Students or your Class Dean formally make such a request to me and suggest a rescheduled due date. You should also take this route if you need an extension but you don't want to tell me why (say, it's for personal or legal reasons). If you explain your reason to a Dean and the Dean tells me it's OK, that's good enough for me.]

• Exams: There will be two mid-term exams and a final, and these will serve as the primary evaluative mechanisms for the course. The exams will be closed-book, and all course material (lectures, handouts, text readings and homeworks) are fair game for the exam. I'll give you relevant formulae, so the exams will be not be memorization exercises. I tend to like problems that ask that you derive results utilizing key derivations from class, then use the result to do something new.

• Extensions: In the interest of fairness I will hold fast to the deadlines and policies above. In particular, I won't grant extensions to homeworks on the day they're due, and I won't reschedule the exam for an individual to a different day from the rest of the class. Should circumstances render such rescheduling unavoidable, you'll need to have the Dean of Students or your Class Dean contact me with a formal request for an extension and a suggested rescheduled time. Such matters should be taken care of before the exam if at all possible. If the matter is personal or confidential, the Deans need not provide me with any particulars, only their opinion that the rescheduling is reasonable.

  The College requires that all written work for a course except for a final be submitted by 5 pm on the last day of classes. The physics department takes this deadline seriously. After that day/time, no homework will be accepted.

Statement of Intellectual Responsibility: particulars for this course

• I strongly encourage you to discuss the homework problems with your classmates---discussing and debating the concepts and procedures is a very effective way to learn. However, the final write-ups of your assignments should be your own. For example, the copying homework solutions in part or in whole from other students (current or prior) or from other sources (commercial or otherwise) is unacceptable. I don't mind if you use other texts to learn from, and if the answer to a homework problem is worked as a example in some other text you may utilize it (although you should first try to solve it on your own as best you can). But in that case you should make an explicit reference to the source in your homework, and of course you still need to explain the solution to the problem in your own words. It goes without saying, but I'll say it anyway: tests are written without help.
• If you have any questions about what is or is not acceptable under this policy, please ask us. In all cases, the spirit of the Statement of Intellectual responsibility will take precedence over legalistic convolutions of the text.

How to get the most from this class:

• Come to class awake and on time.

• Do the relevant reading and read through the problems assigned for the week before the lecture (I'll announce the readings and problems in class and post them on the website). Even if you don't understand the readings fully at the time, try to get the broad outline of the material so you'll have some hooks on which to hang the ideas that we talk about in class.

  The roles of lectures and textbooks
  Lecture will not be a regurgitation of the text, a summary of all you need to know for the course, or a how-to guide for the homework. Rather, I'll try go deeper into selected points. In lecture I'll cover material and do demonstrations related to the readings, but I won't feel obliged to be comprehensive in those places where I feel the text is adequate and I may focus only on a few points that I feel are particularly interesting or subtle. You shouldn't expect to understand what's going on without close study of the readings, and you should come to class with questions you have on the readings. Further, after we settle into the semester a bit, I expect the classes will become less lecture-oriented and more participatory; it will be difficult to reap the maximum benefit from that format if you're not sufficiently prepared to fully participate.

• Ask questions: in class, after class, during office hours, in problem sessions, in the hallways. Ask the questions you think you're supposed to know the answer to, the questions you're sure everyone else knows the answer to (they don't), questions about the overarching logic or the minute details, questions about the points or conclusions that weren't made that seem like they should follow. If individual arguments, their interconnection, or the entire point of the lecture don't seem clear to you, ASK. The main benefit to taking a class, especially at a small school like Amherst, is that you can ask questions of the experts.

• Work the problems: Start early. Try all of them. Struggle with them awhile yourself---some will be clear, simple exercises meant to familiarize you with a procedure, others will be meant to stretch your understanding (some will be intentionally vague, so that you have to clarify the picture before solving the problem). Be sure you can state what ideas (mathematical and physical) are being utilized in each problem; even if you still can't solve the problem you're most of the way there. The ability to work problems is not sufficient to claim an understanding of the physics, but it is a necessary step.

  For the problems you can't solve, talk to classmates, attend the problem sessions, or ask me. When you ask me, either try to give you just enough of a hint to get you through, or I'll guide you through the problem with a series of leading questions. I'll never just tell you how to do it. If you run out of time and don't finish the set, start earlier next week. When the solutions come out, look over them right away, before you've forgotten all of the points you were confused about. You think you'll just get clear on it before the next exam, but there's never as much time as you think.

• If things aren't going as you'd like, take an active role in improving your situation. Use all of the resources at your disposal. Step 1 is almost always to come talk to me. If you don't understand what's going on in lecture, the book, or the problems, if you feel like you can't keep up, come talk to me. I can help you, and I can point you to additional resources. Don't hide.

  On the other hand, if you find the class too slow for your liking, if you have questions that you aren't getting answers to, if you'd like more detail, if you are frustrated that we aren't digging deeply enough, if you crave more applications, come talk to me. I'm very happy to provide you with additional materials or explanations that will will stimulate you and challenge you at whatever level you can handle.

  One word of warning: Amherst College students tend to have lots of extracurriculars of all types. I support this (enthusiastically), and I am occasionally willing to be flexible to facilitate your participation in range of activities, but don't let your extracurriculars overshadow your academics. If you become concerned that your courses are getting in the way of your extracurriculars, you definitely have the wrong mindset. Remember why you're here.

Grading:

• Homework: 20%
• First Midterm: 25%
• Second Midterm: 25%
• Final: 30%

Textbooks:

[This will be the main course text.]

• Classical Dynamics of Particle and Systems (5th edition)
  Author: Stephen Thorton and J. B. Marion (deceased)
  ISBN: 0-534-40896-6
  Publisher: Thomson Brooks-Cole
  

• Classical Mechanics (1st edition)
  Author: John Taylor
  ISBN: 1-891389-22-X
  Publisher: University Science Books
  [This book was used as the course text in the previous three years. I like it, although it's received mixed review from students. At a similar level, but not as dense as Thorton and Marion.]
  
• Physics (vol. 1), Hans Ohanian [a good general text for review of introductory material. used as Physics 23/32 text in the past.]
• Lectures in Physics, R. P. Feynman [Feynman's insights are always unique and worthwhile. You appreciate them more as you get older.]

• Div, Grad, Curl and All That (3rd ed.), H. E. Schey [A usable introduction to vector calculus in a style that physicists find comfortable.]
• Mathematical Methods in the Physical Sciences (3rd ed.), Mary L. Boas [a comprehensive introductory mathematical physics book. very popular around here.]
• Mathematical Handbook (Schaum's Outline Series), Murray R. Spiegel [a convenient compendium of useful mathematical formulae, including integrals and special functions at a low, low price.]
• Basic Training in Mathematics, R. Shankar [textbook for Physics 27]

Key derivations / chains of logic / results to commit to memory

• quick review of Newton's laws and associated mathematics
• curvilinear coordinates and Christoffel symbols
• quick review of projectile motion with friction
• math technique: numerical solutions using mathematica
• quick review: momentum and angular momentum
• quick review: energy
• quick review of harmonic oscillators: damped, driven, resonance phenomena
• math technique: solving second-order differential equations
• math technique: complex numbers
• math technique: integral transforms (Fourier and Laplace)
• Calculus of variations
• math technique: partial differential equations
• Lagrangians and Lagrange's equations
• Two-body central force problems and orbits
• Mechanics in noninertial reference frames
• Motion of rigid bodies and tops
• math technique: matrix algebra, vector spaces, eigenvector and values
• Coupled oscillators and Normal Modes
• Symmetries in mechanics (I)
• Nonlinear mechanics and chaos
• Hamiltonian Mechanics
• Glauber variables
• Hamilton-Jacobi theory
• math technique: symplectic geometry
• Collision theory
• Special Relativity (?)
• Continuum mechanics: strings, waves, membranes, and fluids
• math technique: solving partial differential equations

• Intro
  
• Functions
  
• SymbolicManipulation
  
• Plots
  
• Lists and Arrays
  
• Solving Differential Equations Numerically
  
• Complex Numbers
  
• Fourier Analysis
  
• Programming
  
• Statistics
  
• Input-Output Issues
  
• Solving Equations Numerically
  

Lecture Schedule

Week	Notes	Hmwk	Other
1. September 6	Newton's Laws Sept 7: Introduction to the course / Space, time, vectors General course information. Time, space. Vectors and vector operations. [Handouts: Course information sheet] Sept 9: More on vectors: algebra and calculus Expressing vectors in terms of orthonormal basis vectors. Cartesian and polar coordinate systems. Writing polar coords and basis vectors in terms of cartesian coords and basis vectors, and vice-versa. Scalar ("dot") products of vectors: define, geom. interp, observe it's scalar under rotations. Vector ("cross") product: define in terms of cartesian coords, geometric interp., express in terms of Levi-Civita symbol. Differentiation of vectors (def, chain rule). Diff. of polar unit vectors (both algebraic and geometrical).	Read: TM, Chap. 1 (review of scalars, vectors, and rotation matrices), start reading Chap. 2 Problems: TM: 1.13, 1.22, 1.23, 1.25, 1.27, 1.37, 2.3, 2.5
2. September 12	Newton's Laws Sept 13: Vectors / Newton's Laws Differentiating vectors in polar coordinates. Finite rotations are not vectors, inf. ones are. Definitions: reference frames, mass, force. Newton I: defines inertial frames, describes "natural state" of matter. Newton II: forces change momentum. Newton III: for every action there's an equal and oppose reaction. Strong form and weak form. Newton III fails: in relativity (except for contact forces), in E&M (for magnetic forces). Consequence? Conservation of momentum for composite systems hinges on Newton III. Sept 15: Projectile motion: particles in drag Consider only drag effects (neglect lift), keep terms linear and quadratic in speed. Use dimensional analysis to guess the physics that contributes to linear and quadratic drag terms. Reynolds number is approx. f_quad/f_lin. Some numbers and rule of thumb: if you can see it move, it's quadratic drag. Write Newton II for 2D linear drag with gravity. Solve horiz. velocity eqn. Define terminal velocity from vert. velocity eqn.	Read: TM, Chap. 2 Problems: TM: 2.12, 2.13, 2.20, 2.23, 2.33, 2.36, 2.39, 2.50
3. September 19	Motion of particles / Conservation laws Sept 20: Particles in drag Linear drag: solve vertical motion. Range formula with light linear drag using perturbation expansion. Quadratic drag: solve 1D motion exactly with and without gravity. Define terminal velocity in this case. Note qualitative distinctions between quadratic and linear drag cases. Quadratic in 2D: nonlinear coupled equations, solve numerically. Sept 22: Charges in B-fields / Conservation laws Solve motion of charged particle in constant B-field using complex numbers. Conservation laws: linear momentum, angular momentum. Work-energy theorem.	Read: TM, finish Chap. 2, start Chap. 3 Problems: TM: 2.22, 2.42, 2.43, 2.47, 2.49, 2.52, 2.53
4. September 26	Energy Sept 27: Conservative forces and potential energy Finish work-KE thm. Define PE as energy of position: U[r]= -(work[r0 ->r]). Conservative forces are forces with a PE. Path independence: Two necessary and sufficient conditions--(1) Force has position dependence only (F=F(r)), (2) work is path-independent. Example: gravitational PE from grav. force. Change in U is independent of reference point. E=T+U (total mechanical energy) conserved when only conservative forces act. More generally, change in E = nonconservative work. F is negative gradient of U. Path ind. of F implies circulation of F=0. Local formulation of path independence: Line int. of F is path-ind. IFF curlF=0 everywhere. Forward implication is easy. Backward implication by stokes thm. Sketch proof of stokes thm. Sept 29: Conservative forces / Solving 1D motion using energy Finish proof that curlF=0 iff path integrals of F are path independent. Calculate form of curlF explicitly in cartesian coords. curlF=0 iff F=-grad(U) for some scalar function U. Energy bookkeeping when F=F(r,t). Energy in 1D systems: (1) F is a function of position only implies path independence of work. (2) Graphs of potential energy vs. x are useful in understanding qualitatively the particle motion.	Read: TM, Chap. 3 Problems: TM: 3.6, 3.7
5. October 3	Linear harmonic oscillators I: free and damped SHOs Oct 4: Undamped harmonic oscillators SHO behavior is generic near an equilibrium point, for small enough disturbance. Newton II gives 1D harmonic oscillator equation. Various equivalent forms of solution to equation of motion, including sine and cos, shifted cos, and complex exponential. Show conservation of energy. 2D oscillator (springs in small displacement limit). Isotropic case has general motions which are ellipses. Non-isotropic case gives Lissajous figures or space-filling curves. Oct 6: Damped SHOs Phase diagram for 1D undamped SHO. Lines don't cross on phase plots, by uniqueness. Damped oscillator with linear damping: general solution in terms of exponentials. Three cases: underdamped, overdamped, critical damping. Qualitative features and phase diagram of underdamped and overdamped cases. Form of critically damped solution is slightly different. Brief intro to sinusoidal driving forces.	Read: TM, Finish Chap. 3 / Start Chap. 5 Problems: TM: 3.21, 3.24, 3.26, 3.31
6. October 10	Linear harmonic oscillators II: Driven damped SHOs Oct 11: Fall break Oct 13: Driven damped SHOs Newton II for driven damped SHO. General form of solution for 2nd order linear ODEs: complementary and particlar solutions. Sinusoidal driving force: solve with (complex) exponentials. Express particular solution in terms of amplitude and phase shift. Complementary functions give transients, particular solutions give steady-state behavior. Amplitude resonance: find peak of amplitude as a function of driving frequency. Examine shape of amplitude and phase curves for large and small damping: amplitude is peaked, phase shifts through 90 degrees passing through resonance.	Read: TM, Finish Chap. 3 / Start Chap. 5 Problems: TM: 3.28, 3.29, 3.38, 5.7, 5.9, 5.10	Exam practice: P43F04 exam 1
7. October 17	Linear harmonic oscillators / Matters of gravity Oct 18: Linear harmonic oscillators III: Superposition gone wild Linear superposition. Fourier's theorem. Using Fourier's theorem to obtain response of oscillator to arbitrary driving force. Orthogonality relations. Fourier integrals. Fourier transform pairs. Green's functions with causal boundary conditions. Using Green's functions to obtain response of oscillator to arbitrary driving force. Oct 20: Matters of gravity Newton's law of gravitation for point particles. Newtonian gravity obeys linear superposition. Force due to arbitrary mass distribution. Gravitational field and potential. Shell theorem. Gauss's law for gravitation (integral and differential forms). Poisson's equation for gravitation.	Read: TM, Chap. 6, start Chap. 7 Problems: TM: 6.4, 6.6, 6.8, 6.9, 6.12, 6.15
8. October 24	Variational principles Oct 25: Extremum problems and Euler's equation General intro. Canonical examples: (1) Shortest distance between two points on a plane, (2) Fermat's principle. General variational problem: given a functional, which path extremizes it (assuming fixed endpoints for the path)? Ans: the path described by the Euler equations. Solve shortest distance problem using Euler equations. Oct 27: Euler's equation applied Recap of last time. Show that Euler equations must be zero only if first derivative of the functional vanishes for ALL variations. Shortest path between two points is a line. Brachistochrone problem.	Read: TM, Chap. 7 Problems: TM: 7.4, 7.7, 7.9, 7.15, 7.18, 7.21, 7.25, 7.40 Solutions (.pdf): TM 7.4 TM 7.7 TM 7.9 TM 7.15 TM 7.18 TM 7.21 TM 7.25 TM 7.40
9. October 31	Euler's equation Nov 1: Euler in several variables: with and without constraints Euler's equation with several independent variables. Euler with several variables and a constraint: method of Lagrange multipliers. Applications to mechanics: Define the Lagrangian. Lagrange's equations are equivalent to Newton's second law. They have the form of Euler's equation and can be deduced by applying a variational principle to the action (Hamilton's principle). Lagrange's equations are often simple than Newton, especially in curvilinear coordinates. Nov 3: Lagrange's equation Lagrange's equations have form (generalized force) = d/dt (generalized momentum). Application: 2D motion in central force field, in cartesian and polar coordinates, comparing solutions using Lagrange equations and Newton II. Natural choice of coords gives an analog of the second law which involves the natural analogs of force and momentum. Finding conserved quantities facilitated by Lagrange equation: if L is no explicitly dependent on one of the generalized coords, the corresponding gen. force is zero and the gen. mom. is conserved This is tied to a symmetry of the system related to the generalized coord: L is invariant under translations of the gen. coord. that is missing (Noether's thm). Systems of multiple unconstrained particles are similar: 3N Lagrange eqns corresponding to 3N coords needed to specify the positions of the particles in the system completely. Constrained systems: plane pendulum can be formulated in terms of single variable (polar coords) or two variables with constraint (cartesian coords). Degree of freedom: number of coords that can be independently varied in a small displacement. Holonomic system: # of DOF = # of generalized coordinate. Nonholonomic system requires more generalized coords than there are DOF to specify state of system (e.g. ball rolling on table has 2 DOF, requires 5 gen. coords).	Read: TM, Chap. 8 Problems: TM: 7.23, 7.26, 7.37 Solutions (.pdf): TM 7.23 TM 7.26 TM 7.37
10. November 7	Conservation laws / Hamiltonian mechanics / Two-body central force problems Nov 8: Conservation laws / Hamiltonian mechanics Nov 10: Two-body central force problem	Read: TM, Chap. 9 Problems: TM: 8.8, 8.11, 8.22, 8.32, 9.4, 9.13, 9.56, 9.61 Solutions (.pdf files): TM 8.8 TM 8.11 TM 8.22 TM 8.32 TM 9.4 TM 9.13 TM 9.56 TM 9.61
11. November 14	Two-body central force problem / Dynamics of systems of particles Nov 15: Two-body central force problem Nov 17: Dynamics of systems of particles	Read: TM, Chap. 10, start Chap. 11 Problems: TM: 10.6, 10.8, 10.12, 10.18 Solutions (.pdf files): TM 10.6 TM 10.8 TM 10.12 TM 10.18	Exam practice: P43F04 exam 2
12. November 28	Non-inertial reference frames Nov 29: Nov 29 (evening): Exam 2 Dec 1:	Read: TM, Chap. 11, start Chap. 12 Problems: TM: 11.3, 11.13, 11.17, 11.24, 11.27, 11.28 Solutions: tm11.3 tm11.13 tm11.17 tm11.24 tm11.27 tm11.28	P43F05 exam 2 P43F05 exam 2 problem 2 solution P43F05 exam 2 problem 3 solution P43F05 exam 2 problem 5 solution
13. December 5	Rigid body rotation Dec 6: Dec 9:	Read: TM, Chap. Problems: TM:
14. December 12	Coupled oscillations Dec 13: Dec 15: Reading period	Read: TM, Chap. Problems: TM:

Interesting talks in the Five-College area:

Area Seminars and colloquia

Interesting and useful papers:

Interesting and useful websites:

• -REU, NSF's Research Experience for Undergraduates website