311: Classical Mechanics
Syllabus
Term: Spring
2001
Instructor: John Collins
Text: Classical Dynamics,
Marion and Thornton, 4th edition
Topics:
1. Newtonian
Mechanics of a Single Particle:
This is the same old stuff you did in introductory physics, only
the math is harder. Most
problems lead to differential equations, which are familiar to only some
of you. Maple will help us
solve some of these. Hard
work will pay off.
2. Oscillations: These
are everywhere, and this topic is perhaps the most important of the semester, at
least in terms of its relevance to other branches attention
- we¹ll go through the same mathematics next year.
3. Nonlinear
Oscillations: We won't spend a lot of time here., but since most systems
are nonlinear, at least at large amplitudes, it is worthwhile to examine
some of them. Approximations, phase diagrams, and the transition from
stable to chaotic motion will all be included.
4. Gravitation: This is a prelim to the general topic of a central force that will be discussed later in greater detail. The most important concept developed here is that of potential. Those who have taken E&M should recognize some of this. Those who have yet to take E&M should pay extra attention - we¹ll go through the same mathematics next year.
5. Calculus of Variations: Instead of Newton¹s Laws, it is possible to formulate the laws of physics in terms a minimization principle; e.g. Fermat¹s Final Exam: cumulative, self-scheduled
6. Lagrangian and Hamiltonian Dynamics: Using the math developed in Chapter 5, two new formulations of the laws of physics will be presented: Hamilton¹s Equations and Lagrange¹s Equation. They are, of course, equivalent to Newton¹s Laws, but their advantage lies in the fact that they can make solving problems somewhat easier, and that constants of motion appear quite naturally. (The Hamiltonian is also used extensively in Quantum Mechanics, and the Lagrangian in QED.)
7. Central Force Motion: Orbital motion of the planets about the sun is the most common example of a central force, but more than the 1/r2 force law is considered here; any force F(r) (as opposed to F(r)) is relevant to this section. The concept of the reduced mass helps us simplify the two-body problem to a one-body problem. The conditions for elliptical, parabolic and hyperbolic orbits are derived.
8. Dynamics of Systems with many Particles: (1st half of Chapter 9) This is the description of the motion of an extended body, as opposed to a point particle. The concept of center of mass is critical for this discussion, as the motion is divided into the motion of the center of mass and the motion about the center of mass.
9. Motion in a Noninertial Reference Frame: The spinning of the Earth on its axis means we are not in a valid inertial reference frame. For some situations, it is necessary to include this effect. The centrifugal and Coriolis forces are so-callled quasi-forces that result from such a treatment.
10. Coupled Oscillations: A topic with many application to solid state physics, atomic systems, electrical circuits, and molecular motion. The general formalism is developed here (a little linear algebra is required), and include the concepts of eigenvectors and normal coordinates. Some simple electrical and mechanical systems are discussed.
Grading:
2 take-home tests: 50%
Final Exam (3 hours, open book) 25%
Homework, class participation 25%
1st test: topics 1 - 3 approx. date: first week in March
2nd test: topics 4 - 7 approx. date: second week of April
Final Exam: cumulative self-scheduled