Teaching Fall 2018: Analytic Mechanics

(Physics 511)

INSTRUCTOR: Professor Fulvio Melia OFFICE HOURS: MW 10:00 am - 12:00 am in PAS 447 and most other times, but call 621-9651 or e-mail (fmelia@email.arizona.edu) first to make sure I'm in LECTURES: Tuesday and Thursday, 3:30 pm - 4:45 pm in Chavez Rm 103 HOMEWORK PROBLEMS: Homework # 1: Goldstein: 1.12, 1.2, 1.6, 1.20, 1.22 Homework # 2: Handout (PDF) Homework # 3: Goldstein: 3.33, 3.13, 3.14, Handout (PDF) Homework # 4: Goldstein: 8.14, 8.20, 8.27 Homework # 5: Goldstein: 9.11, 9.25, 9.15, 9.32 HOMEWORK SOLUTIONS: Solutions # 1 Solutions # 2 Solutions # 3 Solutions # 4 Solutions # 5 ABSENCE and CLASS PARTICIPATION: As graduate students, you are not required to attend every lecture, but past experience has shown that engaging with the material in class will significantly improve your understanding of it. In addition, we intend to have discussion along with the lectures themselves, so invariably new questions and ideas are introduced that you may not find in the book or the notes. GENERAL POLICIES: [1] You are all taking this class because you have a passion for this subject. I am not concerned about the possibility that we will encounter any disruption, but we are required by the UA to emphasize our commitment to providing a positive learning experience for everyone. So please do not use any electronic devices (computer or phone) while in class, and let us all be supportive of our fellow students. [2] Please be aware of the UA Threatening Behavior by Students policy, which prohibits threats of physical harm to any member of the University community: policy.arizona.edu/education-and-student-affairs/threatening-behavior-students [3] You must also be aware of the Student Code of Academic Integrity that prohibits plagiarism: deanofstudents.arizona.edu/policies-and-codes/code-academic-integrity [4] You are expected to abide by UA Policy 200E on prohibited behaviors: policy.arizona.edu/human-resources/nondiscrimination-and-anti-harassment-policy [5] If you have a disability, please be aware of the reasonable accommodations provided by the Disability Resources Center: drc.arizona.edu/instructors/syllabus-statement [6] We will try to adhere to the schedule of topics listed below. However, please note that the information contained in this course syllabus, other than the grade and absence policies, may be subject to change with reasonable advance notice, as deemed appropriate by the instructor. TEXTBOOKS: Goldstein, H. et al., Classical Mechanics, Third Edition TOPICS COVERED DURING THE SEMESTER: Survey of the Elementary Principles Mechanics of a particle Mechanics of a system of particles Constraints D'Alembert's Principle and Lagrange's Equations Simple applications of the Lagrangian Formulation Variational Principles and Lagrange's Equations Hamilton's Principle Some techniques of the calculus of variations Derivation of Lagrange's equations from Hamilton's Principle Extension of Hamilton's Principle to Nonholonomic Systems Advantages of a variational Principle formulation Conservation Theorems and Symmetry Properties The Two-Body Central Force Problem Reduction to the equivalent one-body problem The equations of motion and first integrals The equivalent one-dimensional problem, and classification of orbits The Virial theorem The differential equation for the orbit, and integrable power-law potentials Conditions for closed orbits (Bertrand's Theorem) The Kepler problem: Inverse square law of force The motion in time in the Kepler problem Scattering in a central force field Transformation of the scattering problem to laboratory coordinates The Kinematics of Rigid Body Motion The independent coordinates of a rigid body Orthogonal transformations Formal properties of the transformation matrix The Euler angles Euler's Theorem on the motion of a ridig body Finite rotations Infinitesimal rotations Rate of change of a vector The Coriolis force The Rigid Body Equations of Motion Angular Momentum and kinetic energy of motion about a point Tensors and Dyadics The inertia tensor and the moment of inertia The eigenvalues of the inertia tensor and the principal axis transformation Torque-free motion of a rigid body The heavy symmetrical top with one point fixed Small Oscillations Formulation of the problem The eigenvalue of the equation and the principal axis transformation Frequencies of free vibration, and normal coordinates Free vibrations of a linear triatomic molecule Forced vibrations and the effect of dissipative forces The Hamilton Equation of Motion Legendre Transformations and the Hamilton equation of motion Cyclic coordinates and conservation theoremsRouth's procedure and oscillations about steady motion Derivation of Hamilton's equations from a variational principle The principle of least action Canonical Transformations The equations of canonical transformation Examples of canonical transformations The symplectic approach to canonical transformations Poisson brackets and other canonical invariants Equations of motion, infinitesimal canonical transformations, and conservation theorems in the Poisson bracket formulation Symmetry groups of mechanical systems Liouville's theorem Hamilton-Jacobi Theory The Hamilton-Jacobi equation for Hamilton's Principle function The harmonic oscillator problem as an example of the Hamilton-Jacobi method The Hamilton-Jacobi equation for Hamilton's characteristic function Separation of variables in the Hamilton-Jacobi equation Action-angle variables in systems of one degree of freedom The Kepler problem in action-angle variables Hamilton-Jacobi theory, geometrical optics, and wave mechanics PROBLEM SCHEDULE Sections I & II, Elementary Principles (due Sep 18) Goldstein: 1.12, 1.2, 1.6, 1.20, 1.22 Section II (cont.), Variational Principles (due Oct 4) Five problems handed out in class Section III, The Two-Body Central Force Problem (due Oct 30) Goldstein: 3.33, 3.13, 3.14, and one problem handed out in class Sections IV, The Hamilton Equations of Motion (due Nov 15) Goldstein: 8.14, 8.20, 8.27 Section V, Canonical Transformations (due Dec 4) Goldstein: 9.11, 9.25, 9.15, 9.32 METHOD OF EVALUATION Problems (30%) First written, hour-long exam on Tuesday, October 9 (15%) Second written, hour-long exam on Tuesday, November 20 (15%) Written Final, 3:30 pm - 5:30 pm Monday, December 10 (40%)