Physics 43 (Dynamics) Home Page, Fall 2004

Physics 43: Dynamics


11/18/04 I've added the next problem set, and I've added some of what I of the key derivations of Chap 5-9 that you should commit to memory in the Key Derivations section below. You don't always need to memorize the details, just the general framework to allow you to work the details as needed. I don't guarantee the list is complete, so if there are other results you're wondering about feel free to ask me.

11/18/04 We'll have the second midterm exam on Monday Nov. 29, 7-10 pm. The exam will focus on Chapters 6-9 of Taylor (and the part of Chap 5 that wasn't covered on the last exam), although material from earlier chapters is fair game too (I won't focus on it, but of course classical mechanics is cumulative). The problem set for next week will be due on Dec. 2, 11:59 pm.

10/12/04 We currently have the first midterm exam scheduled for Thursday night. I had hoped tho have all of your problem sets back to you before the exam so that you'd have them to study with. Unfortunately, I just heard from our grader Ben that he's been sick over the break and will not be able to finish grading them by exam time. Since I would like you to have your problem sets to look at as a study aid, I'd like to consider delaying the exam a bit. Would you be willing to take the exam on Tuesday evening instead (nominally starting, say, at 7 pm, but with flexibility on the start time for those of you that have prior commitments)? I realize that at this late date such a rescheduling may not be possible or desirable. Please let me know your constraints as soon as possible. When I have heard from most or all of you, I'll get back to you with an update. If I can't establish that next Tuesday is OK, we'll keep the exam as-is, and you should assume the exam will be on Thursday night unless you hear from me otherwise.

09/28/04 I've moved the Problem Set 3 due date from 11:59 pm 9/28 to 11:59 pm 9/30.

09/01/04 I've included an introduction to mathematica below.

08/03/04 I've posted the textbook for the course below.


Course Information

Course Catalog Description:

This course begins with the foundation of classical mechanics as formulated in Newton's Laws of Motion. We then use Hamilton's Principle of Least Action to arrive at an alternative formulation of mechanics in which the equations of motion are derived from energies rather than forces. This Lagrangian formulation has many virtues, among them a deeper insight into the connection between symmetries and conservation laws. From the Lagrangian formulation we will move to the Hamiltonian formulation and the discussion of dynamics in phase space, exploring various avenues for the transition from the classical to the quantum theory. We will study motion in a central force field, the derivation of Kepler's laws of planetary motion from Newton's law of gravity two-body collisions, and physics in non-inertial reference frames. Other topics may include the dynamics of driven, damped oscillators, and non-linear dynamics of chaotic systems. Three class hours per week. Requisite: Physics 27 or consent of the instructor.


Times and places:


Physics 27 or consent of the instructor

Course requirements

Statement of Intellectual Responsibility: particulars for this course

How to get the most from this class:



Required (should be available at the Jeffrey Amherst bookstore): I've asked the library to order a copy of this book. It'll be placed on reserve when it arrives.

Additional useful references (if the library doesn't have them, I'll try to get them):

On mechanics in general: Math books:

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

Tentative Syllabus
Mathematica Tutorials

We will use Mathematica 5.0 fairly extensively in the homework, to obtain numerical solutions to problems that are not analytically solvable and to simplify plotting of results. If you've never used Mathematica before, or haven't used it much, the tutorials will help you get started. They were written by Professor Hilborn and revised by Rebecca Erwin '02. If you download the file and save it to the desktop with a .nb suffix in the name, your computer will recognize it as a Mathematica notebook and will start up Mathematica automatically when you double-click on the icon, provided you have Mathematica installed. Mathematica is installed on lots of the college's public machines, including on the computers in the Physics Department computer lab. Alternately, you can pay the $140 or so to buy the student version.

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.
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.
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

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)


I'll keep scheduling information on this site primarily. I may occasionally use Blackboard as well.

Interesting talks in the Five-College area:

You should start attending the departmental colloquia early and often. They are intended primarily for you, to broaden your exposure to current physics in ways that the department faculty alone cannot. They'll give you an overview of what exciting work is going on in physics and who's doing it. In the beginning you won't always understand all of the talks, but you'll be surprised by how much you can understand even now. In addition, the colloquium food here is better than anywhere else I've ever been. Plus, I organize the colloquia, and it warms my heart to see you there.

Area Seminars and colloquia

Interesting and useful papers:

Interesting and useful websites: