Advanced Analytical Mechanics 553a, b

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1 Lecture Notes for Advanced Analytical Mechanics 553a, b University of Southern California Richard R. Auelmann February 1970

2 About These Notes I was a new engineer at Missile Division of North American Aviation when Sputnik was launched in October My supervisor asked me to give a series of lunchtime lectures on orbital mechanics, since I was familiar with Goldstein s Classical Mechanics. The University of Southern California had a two-semester graduate course Advanced Analytical Mechanics within the Department of Aeronautical Engineering. It so happened that the lecturer scheduled to give the course in September 1959 left unexpectedly. I was asked to step in and ended up giving the course for three hours every Tuesday night for the next 13 years. I started with Goldstein s Classical Mechanics but transitioned into my own notes as time went on. By 1972, I was travelling so much that it was no longer possible to give the course. I never got around to cleaning up the notes for publication. In the process of giving these lectures I learned far more about dynamics and applied mathematics than I ever learned in school. In that sense it was a labor of love. Dick Auelmann

3 1. Dynamics of a System of Particles 1.1 Physical Concepts 1.2 Lagrangian Formulation 1.3 Holonomic Constraints 1.4 Applied Forces 1.5 Energy and Momentum Integrals 1.6 Routhian Formulation 1.7 Liouville Integrals 1.8 Linear Systems Contents 2. Problems in Particle Mechanics 2.1 Kepler Problem 2.2 Problem of Two Fixed Centers of Gravitation 2.3 Restricted Problem of Three Bodies 2.4 Introduction to Hill s Lunar Theory 2.5 Charged Particle in the Field of a Magnetic Dipole 2.6 Motion of a Projectile Through the Atmosphere 2.7 Variable Mass System 3. Dynamics of Rigid Bodies 3.1 Introduction 3.2 Kinematics 3.3 Inertia Matrix and Principal Axis Transformation 3.4 Euler s Equations 3.5 Lagrange s Equations 3. Dynamics of Rigid Bodies 3.1 Introduction 3.2 Kinematics 3.3 Inertia Matrix and Principal Axis Transformation 3.4 Euler s Equations 4. Problems in Rigid Body Mechanics 4.1 Introduction 4.2 Euler Top 4.3 Lagrange Top 4.4 Motion of a Shell in a Uniform Atmosphere 4.5 Motion in a Variable Density Atmosphere 4.6 Libration of a Symmetrical Satellite in a Circular Orbit 4.7 Torques Independent of Body Orientation 4.8 Tippe Top

4 5. Rigid Bodies With Moving Parts 5.1 Quasi Coordinates 5.2 Some Examples 5.3 Satellite Nutation Damper 5.4 Free Motion of a Gyro in a Cardon Suspension 5.5 Pendulum Gyrocompass 5.6 Directional Control of an Automobile 6. Hamilton s Equations and Hamilton-Jacobi Theory 6.1 Overview 6.2 Hamilton s Equations 6.3 Hamilton s Principle 6.4 Canonocal Transformations 6.5 Noncanonical Transformations 6.6 Hamilton-Jacobi Theory 6.7 Separable Systems 6.8 Problem of Two Fixed Centers of Gravitation, Again 6.9 Derivation of Lambert s Theorem 6.10 Method of Variation of Parameters 7. Satellite Motion About an Oblate Spheroid 7.1 Introduction 7.2 Canoncial Constants for the kepler Problem 7.3 Secular Variations of the Kepler Constants 7.4 Method of Sterne and Garfinkel 7.5 Vinti s Method

5 Bibliography 1. Ames, J. S. and Murnagham, F. D., Theoretical Physics, Boston: Ginn & Company, 1929 (New York, Dover Publications, 1957). 2. Birkhoff, G. D., Dynamical Systems, Providence: American Mathematical Society, Chetayev, N. G., The Stability of Motion, New York: Pergamon Press, Inc., Corben, H. C., and Stehle, P., Classical Mechanic, 2 nd ed., New York: Wiley Goldstein, H., Classical Mechanics, Cambridge Mass.: Addison Wesley, Hertz, H., The Principles of Mechanics, orig. German ed 1894, (New York: Dover Publications, 1936.) 7. Lanczos, C., The Variational Principles of Mechanics, Toronto: Univ. of Toronto Press, Lasalle, J., and Lefschetz, S., Stability by Liapunov s Direct Method, New York: Academic Press, Minorsky, N., Nonlinear Oscillations, Princeton, New Jersey: D, Van Nostrand Co., Synge, J. L., Classical Dynamics, Encyclopedia of Physics, Vol III/1: Principles of Classical Mechanics and Field Theory, Berlin: Springer Verlag, Synge, J. L., Tensor Calculus, Toronto: Univ. of Toronto Press, Whittaker, E. T., A Treatise on the Analytical Dynamics of Particles and Rigid Bodies, 4 th ed,, Cambridge: Cambridge University Press, Wintner, A., The Analytical Foundations of Celestial Mechanics, Princeton. New Jersey: Princeton University Press, Yourgrau, W., and Mandelstam, S., Variational Principles in Dynamics and Quantum Theory, 2 nd ed., New York: Pitman Publishing, Thomson, W., and Tait, P. G., Principles of Mechanics and Dynamics, New York: Dover 1962.

6 16. Golubev, V. V., Lectures on the Integration of the Equations of Motion of a Rigid Body About a Fixed Point, Moscow:State Publishing House of Theoretical Technical Literature, (Translation, U. S. Department of Commerce) 17 Bogoliuboff, and Mitropolsky, Asymptotic Methods in the Theory of Nonlinear Oscillations, (translated from Russian) Dehli: Hindustan Publishing Co., Hahn, W., Theory and Application of Liapunov s Direct Method, (translated form German) New Jersey: Prentice Hall, 1963.

PH 610/710-2A: Advanced Classical Mechanics I. Fall Semester 2007

PH 610/710-2A: Advanced Classical Mechanics I Fall Semester 2007 Time and location: Tuesdays & Thursdays 8:00am 9:15am (EB 144) Instructor and office hours: Dr. Renato Camata, camata@uab.edu CH 306, (205)