Physics 43 (Dynamics) Home Page, Fall 2005

Physics 43: Dynamics



I've filled out the remaining problems in problem set due 10/20/05.


I have added a link to last year's exam 1 in the couse outline table below.


Copies of the solutions to the last two problem sets are sitting on the bench outside my office.


For the exam formulas sheet, I'll give you a copy of Appendices D, E and F from Thornton and Marion.


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 5.2 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 / 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


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:

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: