||Lecture, three hours. Kinematics, variational principles and Lagrange equations, rotational dynamics. Hamilton equations of motion, linear and nonlinear perturbation theory, applications to solar system. S/U or letter grading.
||12:20 AM - 1:20 AM
Earth, Planetary, and Space Sciences 201
Classical Mechanics Fall 2017 Syllabus
Classical mechanics will be given this winter quarter by Professor Newman (4640 Geology Building, x5-3912) on Mondays and Wednesdays from 10:00 to 11:50 a.m. in 5655 Geology, beginning Monday, October 4, 2017. This course will focus on developing a thorough understanding of the methods and applications of classical mechanics with an emphasis on nonlinear dynamical principles (such as those emerging in orbital mechanics) and practical problem-solving. Applications will be considered from planetary physics, space plasma physics, and geophysics.
Lecture material will come from the instructor's notes as well as the textbook. The required textbook for the course is Goldstein, Safko, and Poole’s Classical Mechanics, 3rd edition —see reference list appended for these and other useful references. I prefer to use significant material from my notes but will also refer to the text. Some of the most elegant examples of regular and chaotic behavior can be found in the planetary and space environments, but are rarely treated in conventional textbooks. Many such examples will be introduced during the course of the lectures. The course will have 4–5 problem sets assigned at approximately 10 day intervals plus a one-hour mid-term on Wednesday, November 1 and a three-hour final examination (Friday, December 8). Discussion of problem sets among students is encouraged, to the extent that it promotes increased understanding of the material. (Copying is strictly forbidden.) During examinations, an 8 1/2" × 11" formula 2 sheet is permitted. Letter grades will be assigned, unless you have enrolled S/U (note, though, that letter grades are required by those of you needing to satisfy G&SP funda- mental physics requirements). There will be 4–5 problem sets will constitute 60% of the grade and the mid-term examination will be worth 10%; the remaining 30 %points will be assigned to the final examination. My office hours normally will be Monday at 4:00 and Thursday at 1:00 as well as by appointment; students are encouraged to ask questions in and out of class and to seek clarification of the material as it is presented. Auditing will not be permitted. Classical mechanics has always been a staple of physics curricula and is of fundamental importance to the understanding of the dynamical evolution of the solar system and the Earth. Although little will be said in the context of this course, classical mechanics has had a vital historical role in shaping our view of the universe. Three centuries ago, classical mechanics gave rise to seemingly intractable philosophical problems germane to the issue of determinism vs. free-will. As classical mechanics gave way to quantum mechanics, this question was apparently resolved by the advent of the “uncertainty principle,” and classical mechanics was often relegated to the role of helping introduce students to quantum mechanics. In 1954, Kolmogorov, Arnold and Moser discovered an amazing aspect of classical dynamical systems which argued for the existence of a continuum of behavior ranging from regular to chaotic. This opened the door to questions having to do with fractal dimensionality, chaotic behavior, strange attractors, and coherent structures and resulted in a resurgence in popularity for classical mechanics. The Earth, planetary, and space environments are rich in examples of this dynamical complexity—and will provide a pedagogically useful and significant basis for exploring the subject of classical mechanics.
The order of lecture material follows. Unit citations correspond to Goldstein's Classical Mechanics (second edition). Basically, each of the ten units to be covered (relativistic and continuum mechanics are omitted) will take one week (with some allowance for the complexity of the subject material).
- Unit 1: Survey of the Elementary Principles. Supplementary material on tensor notation and on variational principles and geometry.
- Unit 2: Variational Principles and Lagrange's Equations. Supplementary material on the role of integral invariants, integrability, degrees of freedom, and embedding dimension.
- Unit 3: The Two-Body Central Force Problem. Supplementary material on symmetry breaking, bifurcations, the role of angular momentum, and the restricted three body problem.
- Unit 4: The Kinematics of Rigid Body Motion. Supplementary material on the Jacobi integral, group theory, the rotation group, and the geometry of rotation.
- Unit 5: The Rigid Body Equations of Motion. Supplementary material on the geometrical complexity of rotating tops.
- Unit 6: Small Oscillations and Special Relativity. Supplementary material on Fourier analysis, eigenvalue-eigenvector decomposition, stability, and nonlinear perturbation expansions.
- Unit 7: The Hamilton Equations of Motion. Supplementary material on electro-magnetic fields in Hamiltonian mechanics.
- Unit 8: Canonical Transformations. Supplementary material on dynamics in a uniform magnetic field.
- Unit 9: Hamilton-Jacobi Theory. Supplementary material on integrability and chaos, the KAM theorem, and the path to quantum mechanics.
- Unit 10: Canonical Perturbation Theory. Supplementary material on adiabatic invariants and on electromagnetic fields.
- Andronov, A.A., A.A. Vitt, and S.E. Khaikin 1966. Theory of Oscillators (Oxford: Pergamon Press).
- Arnold, V.I. 1989. Mathematical Methods of Classical Mechanics 2nd edition (New York: Springer-Verlag).
- Danby, J.M.A. 1989. Fundamentals of Celestial Mechanics 2nd edition (Richmond, Virginia: Willmann-Bell, Inc.).
- Goldstein, H., C. Poole, and J. Safko 2001. Classical Mechanics 2nd edition (San Francisco: Addison Wesley).
- Guckenheimer, J. and P. Holmes 1983. Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields (New York: Springer-Verlag, Inc.)
- José, J.V. and E.J. Saletan 1998. Classical Dynamics: A Contemporary Approach (Cambridge: Cambridge University Press).
- Lanczos, C. 1970. The Variational Principles of Mechanics 4th edition (Toronto: University of Toronto Press).
- Landau, L.D. and E.M. Lifshitz 1976. Mechanics 3rd edition (Oxford: Pergamon Press).
- Lichtenberg, A.J. and M.A. Lieberman 1983. Regular and Stochastic Motion (New York: Springer-Verlag, Inc.).
- McCauley, J.L. 1997. Classical Mechanics: Transformations, Flows, Integrable and Chaotic Dynamics (Cambridge: Cambridge University Press).
- Pars, L.A. 1979. A Treatise on Analytical Dynamics (Woodbridge, Connecticut: Ox Bow Press).
- Scheck, F. 1990. Mechanics: From Newton's Laws to Deterministic Chaos (Berlin: Springer-Verlag, Inc.).
- Sommerfeld, A. 1942. Mechanics (New York: Academic Press, Inc.).
- Whittaker, E.T. 1937. A Treatise on the Analytical Mechanics of Particles and Rigid Bodies with an Introduction to the Problem of Three Bodies 4th edition (Cambridge:
Cambridge University Press).
- Zaslavsky, G.M., R.Z. Sagdeev, D.A. Usikov, and A.A. Chernikov 1991. Weak Chaos
and Quasi-Regular Patterns (Cambridge: Cambridge University Press).