Physics 706

Theoretical Mechanics

Fall 2009

 

Professor: Lev Kaplan

Lectures: MWF 3 – 3:50, in Boggs 242

Textbooks: Fetter and Walecka, Theoretical Mechanics of Particles and Continua

                  Baker and Gollub, Chaotic Dynamics, an Introduction

Possibly useful reference books: Goldstein, Classical Mechanics (the “standard” text)

                                                Landau and Lifshitz, Mechanics (very concise classic)

                                                 

Office: 5046 Percival Stern Hall

Office hours: F 1:00 – 2:00 (to be confirmed), or other times available by appointment

            Please stop by and say hello – I am always happy to see you

Email: lkaplan@tulane.edu [A great way to ask a question or arrange an appointment]

Telephone: 504-862-3176 (x3176) [Please leave a message if I’m not there or try email]

 

Welcome to Graduate Classical Mechanics at Tulane University!

 

General Course Objectives and Requirements:

 

This is an introductory graduate-level survey of classical mechanics, designed to familiarize you with the various approaches to this subject (including the Newtonian, Lagrangian, and Hamiltonian formulations). Although we begin with Newton’s laws, a solid background in classical mechanics at the intermediate undergraduate level is strongly recommended. Our focus will be on physics rather than on mathematical formalism and rigor, and the needed mathematical tools (e.g., basic linear algebra) will be reviewed as needed. Topics covered will include: Newtonian mechanics (with an emphasis on conservation laws, central forces, and rotating coordinate systems), Lagrangian mechanics and the variational principle, small oscillations and normal modes, anharmonic, damped, and driven oscillations, Hamiltonian dynamics, nonlinear dynamics and chaos, and continuum mechanics. We will take advantage of multiple opportunities to discuss interesting connections between classical and quantum physics. In addition, the course will include a brief introduction to numerical methods (particularly Runge-Kutta integration), and you will be able to apply these methods as an integral part of our study of nonlinear and chaotic systems.

 

Grading:

 

Homework:   30%

In-class midterm exam (around Friday, October 9):  30%

Comprehensive final exam (Monday, Dec 14, 1:00 – 5:00):  40%

 

Homework:

 

Homework will be assigned every one or two weeks. The assignments may include derivations and examples that extend the class discussion. These assignments will be done individually, except as otherwise instructed (though of course I strongly encourage discussions with your colleagues). Please do not hesitate to come to me when you have questions. Late homework will generally be subject to a penalty, unless prior permission is obtained from the instructor (such permission will be granted only in exceptional circumstances).

Examinations:

 

One midterm and a comprehensive final exam will compose 70% of your final grade. The date of the midterm is subject to minor adjustment, but the date of the comprehensive final exam is set by the university. Each exam may contain a combination of definition/conceptual questions (short answer), derivations, and calculations. Exam problems and questions will be drawn from material covered in class as well as from homework problems. More information will be provided prior to the date of each exam.

 

Team Study:

 

Team study is strongly encouraged, and is a habit that will benefit you in all your other courses as well. You are especially encouraged to consult with your colleagues in the class when you encounter difficulties with the coursework, as peer-to-peer learning will enhance your understanding of the material and make study more enjoyable (in those instances where you are the one offering assistance and also in those instances where you are seeking out assistance). Please note that the peer-to-peer learning I recommend is in addition to, and not a substitute for, the assistance that I personally will be more than happy to provide whenever you have questions (during class, after class, during office hours, or by appointment).

 

Tentative Course Outline:

 

This outline is subject to likely adjustments, which will be announced in class.

 

1. Newtonian Mechanics (Fetter & Walecka chapters 1, 2)

            Newton’s Laws and Conservation Laws

            Systems of Particles and Center-of-Mass Coordinates

            Central Forces, Effective Potentials, and Orbits

            Example: Gravitational Potential

Scattering and Cross Sections

            Accelerating and Rotating Coordinate Systems

            Example: Motion on Earth’s Surface

 

2. Lagrangian Dynamics (F & W chapter 3)

            Constrained Motion

            Lagrange’s Equations and D’Alembert’s Principle

            Variational Principle

            Symmetries of the Lagrangian and Conservation Laws

            Action Function and Quantum Mechanics

            Forces of Constraint

 

3. Small Oscillations (F & W chapter 4)

            Normal Modes

            Review of Linear Algebra

            Analogy with Quantum Wave Functions

            Avoided Level Crossings and Adiabatic Evolution

            Example: Coupled Pendulums

            N-body Problems

            Example: Vibrating String

            Continuum Limit, Dispersion Relations, and Effective Theories

 

4. Anharmonic, Damped, and Driven Oscillations (Landau and Lifshitz)

            Analytic Treatment and Perturbative Expansions

Resonances and Hysteresis

Example: Driven Pendulum

Numerical Methods: Differentiation, Integration, Runge-Kutta

 

5. Hamiltonian Dynamics (F & W chapter 6)

            Hamilton’s Equations

Canonical Transformations

Hamilton’s Principal Function

Hamilton-Jacobi Theory and Action-Angle Variables

Example: Harmonic Oscillator

Poisson Brackets

Transition to Quantum Mechanics: Dirac Brackets and Schrödinger Equation

 

6. Chaotic Systems: Unpredictable Determinism (Baker and Gollub)

            Phase Space Trajectories, Poincaré Sections, and Discrete-Time Maps

            Conservative and Dissipative Systems

Regular and Chaotic Motion

Attractors

Quantum Mechanics

 

7. Continuum Mechanics: Strings and Membranes (F & W chapters 7, 8) [if time permits]

            Classical Field Theory

            Wave Equation

            Fourier Series and Eigenfunction Expansions

            Variational Methods

            Green’s Functions

            Perturbation Theory

            Higher Dimensions

 

About me:

 

My present research interests involve waves in complicated geometries (from ocean waves to electron waves). This field is known as quantum chaos, and if you want to know more about it, just ask, or take a look at http://www.tulane.edu/~lkaplan. I was born in Latvia (former USSR), grew up in New Jersey, did my undergraduate studies at the University of Pennsylvania, and my graduate work at Harvard University (on particle theory). Prior to coming here, I worked on nuclear theory at the University of Washington in Seattle. I strongly encourage you to stop by my office to talk about physics, Tulane, questions, concerns, suggestions, or anything else that may be on your mind.