Instructor

Professor William Loinaz
Office: 223 Merrill Science Center
Phone: (413) 542-7968
email: waloinaz@amherst.edu
Office Hours: MW 3:30-4:30, T 2-3
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: Ben Heidenreich / Amrit Amirapu / Meltem Duran

Course Information

Course Catalog Description:

Schedule

• Class: Tuesday and Thursday 10:00-11:20, Merrill 211
• Problem Session (informal): Mondays at 9pm, 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 Tuesdays @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 Friday @11:59 pm), and will receive no credit after that.

  Why do the homework?
  Since you're juniors and seniors, 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 Mechanics (1st edition)
  Author: John Taylor
  ISBN: 1-891389-22-X
  Publisher: University Science Books

• 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.]

• 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 (2nd 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

• Chap 5: Oscillators
  • Conversion between trig functions and exponentials
  • Solutions of homogeneous second-order linear ODEs with constant coefficients: in terms of trig functions and in terms of exponentials.
  • Inhomogeneous equations: general form of solution, explicit expressions for phase shift, amplitude, resonance parameters in terms of oscillator characteristics and driving force.
  • Fourier series: series expansion in terms of exponentials and trig functions, how to obtain fourier coefficients in the Fourier decomposition of periodic functions, orthogonality relations
  • Same thing for Fourier integrals
  • Use Fourier analysis to solve drive oscillator problem
  • Derivation of Green's function for damped SHO, and it's use in solving for motion of damped driven SHO.
• Chap 6: Calculus of Variations
  • Derivation of Euler-Lagrange equations in absence of constraints (for one or several variables)
• Chap 7: Lagrange equations
  • Show explicitly how to go directly from any one to any other: Newton's 2nd law, Lagrange equation, Hamilton's principle (without constraints)
  • Demonstrate Hamilton's principle in the presence of constaints
  • Demonstrate conservation of momentum, angular momentum, energy starting from a Lagrangian with the appropriate symmetry
  • Show that Lagrange multipliers can be used to incorporate constraints in Lagrangian systems
• Chap 8: Two-body central force problem
  • Show the step-by-step decomposition of the two-particle central force problem down to 1 particle, 1D motion and some other simple conditions.
  • Demonstrate existence of bound and unbound orbits using energy methods
  • Solve Kepler problem (1/r potential) and characterize orbits
  • Derive Kepler's 3 laws
• Chap 9: Non-inertial frames
  • Show the presence of tidal effects on the earth
  • Deduce the form of the time derivative for vectors in rotating frames (with constant angular velocity)
  • Derive Newton's 2nd law in rotating frame
• Chap 10: Rotational motion of rigid bodies
  • Basic kinematic quantities for rigid bodies (ang. mom, CM, energy, etc.)
  • Moment of inertia tensor (def. and calculation)
  • Principal axes and moments (def., calculation, implications of symmetry)
  • Precession of top under gravity: how to derive precession freq.
  • Derivation of Euler equations
  • Intermediate axis thm: derive
  • Euler equations: application to simple cases
  • Euler angles: def., write key quantities in terms of them
  • Spinning top via Lagrangians: how to set it up
• Chap 11: Normal modes and coupled oscillators
  • Set up matrix equations for system of coupled oscillators
  • Find normal frequencies
  • Solve for general motion in terms of normal modes, describe motion
  • Normal coordinates
• Chap 13: Hamiltonian Mechanics
  • Obtain Hamiltonian from Lagrangian
  • Hamilton's equations of motion: derive and use
  • Liouville's Theorem: derivation and result

• quick review of Newton's laws and associated mathematics
• 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 General course information. Time, space. Vectors and vector operations. Reference frames. Mass and force. Newton's Laws I and II. [Handouts: Course information sheet, first problem set ] Sept 9: Projectile Motion with Drag Newton III. Conservation of momentum. Failure of Newton III (e.g. magnetism). Derivatives in non-cartesian coordinates. Motion of particles with air resistance. Linear and quadratic drag. Dimensional analysis to obtain scaling of coefficients for drag terms. [Handouts: none]	Read: Taylor, Chap. 1 and 2 Problems: Taylor: 1.33, 1.34, 1.40, 1.48, 1.50, 2.4, 2.14, 2.21, 2.22, 2.36
2. September 13	Newton's Laws Sept 14: Projectile motion with drag Reynolds number. Solving equations of motion. Linear drag equations can be solved exactly, in 1D or in 2D (with gravity). Range formula: zero drag case exactly, linear drag case (light drage) using perturbation expansion. Quadratic drag: 1D motion can be solved exactly, with and without gravity. 2D motion with gravity must be solved numerically (Only skimmed quadratic case in lecture.) Sept 16: Conservation of linear and angular momentum Linear momentum: Newton II implies conservation of total linear momentum for systems in which external force is zero. Rocket problem by cons. of linear momentum. Center of mass defined. Newton II for system of particles in terms of center of mass. Angular momentum: Definition, for a single particle. Time derivative of L is torque (rotational Newton II). If forces are central, torque about center is zero and L is constant. Constant L simplifies motion from 3D to 2D. Conservation of L implies Kepler II (planets sweep out equal areas in equal times). Derive this result geometrically and by direct differentiation of vectors.	Read: Taylor, Chap. 2 and 3 Problems: Taylor: 1.18, 2.42, 2.44, 2.49, 2.53, 3.11, 3.12, 3.30, 3.34, 3.35
3. September 20	Conservation Laws Sept 21: Angular Momentum of systems of particles / Kinetic energy Angular momentum for systems of particles: Define total L, derive rotational Newton II (i.e. relate total L to torques). Internal torques cancel if internal forces (1) obey Newton III and (2) are central. Then, total L changes only due to external torques, and total L is conserved if net external torque is zero. Moment of inertia defined, relates L and w in rigid rotation about an axis. Rotational Newton II can hold in some circumstances even in accelerated frame. Energy: History and context. Kinetic energy defined. Related to force and work via work-kinetic energy theorem. Potential energy and conservative force defined. Sept 23: Energy Forces are conservative when (1) position dependence only (2) work is path-independent. When all forces are conservative, total mechanical energy E=T+U is conserved. U for conservative force is negative of work done by that force. If not, work by non-conservative forces accounts for loss of E. For conservative forces, F= -(grad U). Path independence iff curl F = 0 (by Stokes thm). Explicitly evaluating curl F in cartesian coordinates.	Read: Taylor, Chap. 3 and 4 Problems: problem set 3
4. September 27	Energy and oscillators Sept 28: Conservative forces and 1D motion Conservative Forces: Finish evaluating curl F in cartesian coordinates. Show that curl F = 0 implies that F is the gradient of a scalar function. Summarize vector calculus identities. Generalize to time-dependent forces. Energy in 1D systems: Path independence is simple in 1D. Graph of potential energy vs. x gives simple means of visualizing motion of a system. Motion can be solved completely by conservation of energy. Sept. 30: Curvilinear 1D motion, central forces, and oscillators Curvilinear 1D systems: Arc length parametrization is natural. Express KE, tangential forces in terms of s. Newton II in terms of s. Central Forces: Central force is conservative iff central force is spherically symmetric. Oscillators: Harmonic oscillator generically describes motion about equilibrium points under small displacments. Write down SHO equation, show solution in terms of phasors.	Read: Taylor, Chap. 4 and 5 Problems: Taylor, 5.4, 5.12, 5.13, 5.18, 5.19, 5.27, 5.32, 5.43, 5.44, 5.45
5. October 4	Oscillators Oct. 5: Phase portraits and damped oscillators Isotropic 2D oscillator. Anisotropic oscillator: Lissajous figures and space-filling curves. Phase diagram for SHO: curves are ellipses. SHO with linear damping (friction). General form of solution, and 3 possible cases. Underdamped oscillator. Oct. 7: Damped and driven oscillators and the principle of superposition Phase diagram for underdamped, overdamped, and critically-damped oscillators. Driven oscillators. General discussion of 2nd order linear inhomogeneous ODEs with constant coefficients: homogeneous and particular solutions. Solving with sinusoidal driving force: phase shift, amplitude, resonance.	Read: Taylor, Chap. 5 Problems:
6. October 11	Harmonic oscillators and the principle of superposition Oct. 12: (Fall break) Oct. 14: Principle of superposition: Fourier analysis and Green's functions Principle of superposition for linear operators. Solution to damped harmonic oscillator with sinusoidal driving force. Fourier's theorem. Fourier expansions of periodic functions, in terms of sin and cos or exponentials. Orthogonality relations. Solution to damped harmonic oscillator with arbitrary periodic driving force. Fourier integrals and Fourier transforms. Solution to damped harmonic oscillator with arbitrary NON-periodic driving force. Green's function as solution to impulsively-driven damped SHO (with causal boundary conditions). Superposition + Green's function provides solution to damped SHO driven by arbitrary driving force.	Read: Taylor, Chap. 6 Problems: Taylor, 5.49, 5.53, 5.56, 6.5, 6.6, 6.17, 6.19, 6.22, 6.23, 6.24 (Due Thursday Oct. 21)
7. October 18	Calculus of variations Oct. 19: Calculus of variations and the Euler-Lagrange equations General intro. Examples extremum problems: shortest path between two points, Fermat's principle. General formulation of extremum problems. Derivation of Euler-Lagrange equations by first variation. Solve shortest path problem. Oct. 19, 7 pm: Midterm 1 Oct. 21: Calculus of variations and mechanics: the Lagrange equations Solve brachistochrone: path is a cycloid. Caution on minimum point vs. stationary point of action. Mention problem of geodesics on a sphere. Application to mechanics: Lagrange equations. Brief discussion of virtues of variational approach. Unconstrained motion of a particle in a conservative field. Define Lagrangian L, show that Newton II can be expressed in terms of derivatives of L (Lagrange equations). Define action S in terms of L. Show that Lagrange equations obtain from stationary condition on S: physical paths of a particle are stationary in S (Hamilton's Principle). Claim that Newton II, Lagrange equations, and Hamilton's Principle are all equivalent (at least for unconstrained systems).	Read: Taylor, Chap. 7 Problems: Taylor, 6.1, 6.14, 6.16, extra
8. October 25	Lagrangian mechanics Oct. 26: Lagrangian mechanics in constrained and unconstrained systems Unconstrained systems: Virtue of Hamilton's principle-- allows easy formulation in any coord sys. Define generalized coords. Example of Lagrange eqns: constrained particle in 2D, using polar coords. "Natural" choices of coords give "natural" analogs of Newton II. Conserved quantities in Lagrange formulation--tied to symmetries (invariance of L under translations of gen. coords). Extension to multiple unconstrained particles. Constrained systems: Example: Plane pendulum as constrained system. Can't vary coords independently because of constraint. Degrees of freedom. Holonomic and non-holonomic constraints. Lagrange equations for holonomic systems. Oct. 28: Constrained systems Prove Hamilton's principle for variations consistent with constraints. Lagrange's equations for constrained systems using "natural" coordinates for constraint surface.	Read: Taylor, Chap. 7 Problems: Taylor, 7.14, 7.20, 7.28, 7.29, 7.35, 7.36, 7.40, 7.41
9. November 1	Lagrangian Mechanics / Two-body central force problems Nov. 2: Conservation laws in Lagrangian mechanics Ignorable coordinates imply conservation laws. L is symmetric under translations of ignorable coordinates. Symmetry under translations of cartesian coords implies conservation of linear mom. Symm. under translation in time implies conservation of energy. Hamiltonian is total energy of system if generalized coords are natural and L is not explicitly time-dependent. Nov. 4: Breaking down the two-body central force problem Write Lagrangian for system. Choose center-of-mass coordinates and show that Lagrangian separates into a pair of decoupled lagrangians, one for a free particle and one for a particle moving under conservative central potential U(r). In center-of-mass frame the free particle Lagranian is trivial. In remaining one-particle problem angular momentum is conserved, so motion is really 2D. Lagrange's equation for polar angle expresses conservation of momentum. Radial equation for effective 1D problem contains term due to real force and term due to centrifugal force. 1D problem can be analyzed using energy methods, including potential energy term due to centrifugal force (centrifugal barrier).	Read: Taylor, Chap. 8 Problems: Taylor, 7.46, 7.47, 7.52, 8.2, 8.11, 8.14, 8.17, 8.20, 8.21, 8.22
10. November 8	Central force motion / Non-inertial reference frames Nov. 9 : Central Force Motion and the Kepler Problem Qualitative features of radial motion from energy. Centrifugal barrier. Orbits close only if periods of radial and angular motion are commensurate, as occurs in inverse-square force law. Obtain equation for r as a function of angle phi, solve for case of inverse square force law. Categorize orbits according to eccentricity. Nov. 11: Kepler Problem / Noninertial Reference Frames (NIRFs) Kepler: Show that orbits are elliptical for ecc. < 1. Brief review of geometry of ellipse. Relate ecc. to energy. Kepler's 3 laws. NIRFs: General discussion. Inertial forces in linearly accelerated frame. General discussion of rotating frames. Statement of Euler's theorem. [Handouts: Goldstein, sec. 4.6; Solutions to PS 8; AJP article on Bertrand's thm (by Lowell Brown)]	Read: Taylor, Chaps 8 and 9 Problems: Taylor, 8.31, 8.34, 9.3, 9.11, 9.14
11. November 15	Non-inertial reference frames Nov. 16: Rotating reference frames Infinitesimal rotations are vectors (although finite rotations are not). Angular velocity vector. Time rate of change of vector attached to rotating body. Addition of relative angular velocities. Time derivatives in rotating frames. Newton's second law in rotating frames. Introduce centrifugal and coriolis forces. Nov. 18: Coriolis and Centrifugal Forces Show that finite rotations do not in general commute, but infinitesimal rotations do. Centrifugal force. Contribution of centrifugal force to effective g. Coriolis force. Effect similar to charged particle motion in magnetic field. Cyclonic motion of air flow. Precession of Foucault pendulum.	Read: Taylor, Chap. 9 and 10 , Problems: Taylor, 9.22, 9.28, 9.30, 9.31, 10.8, 10.13, 10.18, 10.27, 10.31, 10.38
12. November 29	Rigid Body Rotation Nov. 30: Rigid Body Motion: review, and surprises Center of mass, second law, angular momentum, rotational second law, energy for systems of particles. Observe that these generically break into a piece describing COM motion and a piece describing motion about the COM. For a single particle rotating about axis, ang. mom. and ang. vel. are not collinear. Express relationship between ang. mom. and ang. vel. in general. Products and moments of inertia. Dec. 2: Inertia tensor Inertia tensor, in matrix and index forms. I symmetric. Use of symmetries to simplify inertia tensor. Example: cube rotated about corner. Observe that L and w are collinear for rotations about diagonal, NOT about edge. Example: cube rotated about center--symmetries imply I proportional to unit matrix. So, L and w parallel for any rotation. Example: cone rotated about its tip, z-axis as symmetry axis. L and w parallel for w along coordinate axes, not for other axes. Principal axes and principal moments. Finding principal mom. and axes. Symmetries help. Generally, solve eigenvalue problem: since I is real, symm. tensor, can always be diagonalized by a rotation. Eigenvalues are principal moments, eigenvector principal axes, matrix of eigenvectors diagonalizes I.	Read: Taylor, Chap. 10 and 11 Problems: Taylor, 10.33, 10.37, 10.41, 10.43, 10.52, 10.55, 11.5, 11.9, 11.15, 11.19
13. December 6	Rigid Body Rotation and Coupled Oscillations Dec. 7: Symmetric Tops and Euler's Equations Symmetric Spinning Tops: Basic idea: Precession of top in gravitational field (assuming large L). Euler's equations: Space frame vs body frame. Deriving Euler eqn. Special cases: Euler eqn for symmetric tops with torque. Euler eqn in torque-free case for general top. Intermediate axis thm. Dec. 9: Euler's equations and Coupled Oscillators Euler's equations: Finish intermediate axis thm. Solve Euler eqns for symmetric top in absence of torques. Description of precession: body cone, space cone. Coupled oscillators: Two masses, three springs: formulated as matrix problem. Eigenvalues are normal frequencies, all components oscillate with same frequency. Eigenvectors are normal coordinates, they describe the motions of the components in a normal mode.	Read: Taylor, Chap. 11 and 13 Problems: Taylor, 11.24, 11.29, 11.31, 11.34, 13.7, 13.18, 13.23, 13.25 (due Wed. 5 pm)
13. December 13	Coupled Oscillators and Hamiltonian Mechanics Dec. 14: Normal Modes and Hamiltonian Mechanicsn Normal Modes: Two masses, three springs, special case of all m and k equal. Work out normal modes and normal coordinates explicitly. Hamiltonian Mechanics: Virtues of Hamiltonian mechanics. Basic variables: generalized coordinates, velocities, momenta recalled. L, H, configuration space and phase space defined. Deriving H from L in simple case. Legendre transform. Hamilton's equations of motion. Example: Central force problem in Hamiltonian equations. Ignorable coordinates. q,p symmetry of H. Features of phase space: (1) orbits don't cross, (2) Liouville's thm: volume in phase space evolves as incompressible fluid. Dec. 16: (Reading Period)	Read: Taylor, Problems: None
December 21	Final exam: Dec. 21, 9-12 am (Merrill 220)

Website

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
  
• Physics applets from MSU: Physlets
  
• drussell