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.
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.]
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.
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.
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.
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.
Key derivations / chains of logic / results to commit to memory
|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.
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
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.
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.
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
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 force is conservative iff central force is spherically symmetric.
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
|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
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.
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).
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
Finish intermediate axis thm. Solve Euler eqns for symmetric top in absence of torques. Description of precession: body cone, space cone.
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
Two masses, three springs, special case of all m and k equal. Work out normal modes and normal coordinates explicitly.
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)
|December 21||Final exam: Dec. 21, 9-12 am (Merrill 220)
Area Seminars and colloquia