## Physics 281 — Classical Mechanics

#### Objectives and Goals

The core formalisms, concepts, and techniques of classical mechanics are essential components of a graduate education in physics. The subject forms the basis for a thorough understanding of quantum mechanics and statistical mechanics, as well as the foundation for modern pure and applied research on nonlinear mechanical systems.

Newtonian and Lagrangian methods are reviewed and applied extensively to translational and rotational motion of rigid bodies, the central force problem, and small oscillations of mechanical systems with several degrees of freedom. Students should learn to set up the Lagrangian for reasonably complicated systems, analyze the relevant symmetries, reduce the dimensionality of the system where appropriate, obtain the equations of motion, and solve them in certain cases. The Hamiltonian formulation is also presented. Students should develop some facility with the concepts of canonical transformations, phase flow, Poisson brackets, integrable systems and chaos. Students should develop an appreciation for the consistency of the various formulations and their interconnections.

The course provides exposure to the following topics:

• variational principles;
• Lagrangians for simple mechanical systems and for a particle in an electromagnetic field;
• the relation between symmetries and conservation laws;
• motion in a rotating frame (centrifugal and Coriolis forces);
• effective potentials and the angular momentum barrier;
• the Kepler problem (bound and scattering orbits);
• normal modes of linear systems;
• the behavior of "small molecules" (ball-and-spring systems);
• Euler angles, inertia tensors, and the heavy symmetrical top;
• resonance in linear and weakly nonlinear damped oscillators;
• perturbation theory for weakly nonlinear oscillators;
• stability analysis for solutions to equations of motion;
• action-angle variables and adiabatic invariants; and
• Poincare sections and Hamiltonian chaos.

This course also serves to introduce first-year graduate students to the level of mathematical rigor required for PhD level research. Students are presented with difficult problems and required to work through them carefully and thoroughly. Relevant skills that should be developed include performing difficult calculations, examining solutions for physical coherence, and obtaining consistent solutions using different approaches.

### Methods and Approach

Prerequisites:
Undergraduate courses covering the basics of Newtonian mechanics, differential equations, and linear algebra are essential. An undergraduate course in classical mechanics that introduces Lagrangian and Hamiltonian formalisms is recommended. Some familiarity with quantum mechanics and statistical mechanics at the undergraduate level may also be helpful in enabling students to see important connections between these topics and classical mechanics.
Format:
This course is taught through 75-minute lectures (2 per week) and weekly homework sets. Lectures generally involve blackboard presentations by the professor, but student participation is encouraged. Homework sets generally consist of 3-4 problems designed to take approximately 10 hours of concentrated effort to complete. Students are encouraged to discuss the homework with their peers, but are required to write solutions independently. Students are responsible for all material covered in the lectures and in the homework problems. Some concepts and applications that are essential for future courses are covered only in the homework.
Texts:
The course is taught at the level of Goldstein's "Classical Mechanics" or Jos\'{e} and Saletan's "Classical Dynamics." The choice of primary text is left to the instructor.
• Marion and Thornton's "Classical Dynamics" treats much of the course material at an intermediate level;
• Landau and Lifshitz "Mechanics" treats much of the course material with extreme elegance;
• Ott's "Chaos in Dynamical Systems" gives a useful introduction to modern theories of chaos.
There is a final exam for the course and at least one midterm. Exams are designed to test each student's grasp of the fundamental concepts. Each exam contains some problems of the type that appear on the qualifier exam. Grades for the course are determined by homework (50%), midterm(s) (20%), and the final exam (30%).

### Sample Syllabus

(There are two lectures per week.)

• Lecture 1: Newton's laws
• Lecture 2: types of forces, dissipation
• Lecture 3: Lagrangian formalism: variational principles
• Lecture 4: least action, Lagrangian eqns of motion
• Lecture 5: Conjugate variables, phase space
• Lecture 6: symmetries and conservation laws
• Lecture 7: rotating frames of reference
• Lecture 8: Rigid-body rotation: rotation matrices
• Lecture 9: inertia tensors, Euler angles
• Lecture 10: the heavy top
• Lecture 11: Small oscillations: normal modes
• Lecture 12: reduction of complexity using discrete symmetries
• Lecture 13: ordinary resonance
• Lecture 14: parametric resonance
• Lecture 15: weakly nonlinear oscillator
• Lecture 16: Central forces: reduction to 1D; Kepler
• Lecture 17: scattering
• Lecture 18: Hamiltonian formalism: Hamilton's equations
• Lecture 19: canoncial transformations; Liouville's theorem
• Lecture 20: Poisson brackets
• Lecture 21: Hamilton-Jacobi equation; action-angle variables
• Lecture 22: chaos
• Lecture 23: Topic of professor's choosing
• Lecture 24:
• Lecture 25: special topic or review
• Lecture 26:
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