Physics 611
Classical Mechanics
Prof. Bala Sundaram
bala.sundaram@umb.edu
Office/Phone: S3-108 / 7-6055
Class: Tu/Th: 4:30-6:45 p.m., S02-066
Fall 2012
Office hours: Tu/Th 11-12
or by appointment
Course outline: The course will focus on the topics in classical mechanics which form the
foundations for a number of other areas in modern physics encountered in any graduate
curriculum. Along with the more traditional topics such as variational principles, central force
motion and the Lagrangian & Hamiltonian formulations, this class will include an introduction to
some of more recent developments in classical dynamics. In particular, this class will deal with
concepts in nonlinear dynamical systems with their application in areas such as fluid dynamics and
astrophysics. In particular, issues of non-integrability and the associated new ideas including
deterministic chaos will be addressed. Further, the links between canonical perturbation theory and
quantum mechanics will be explored if time permits.
Textbook : There is no official textbook required for the class though we will be drawing on
several texts for covering the course material. Class notes will be made available. There are a
number of canonical texts and some more ‘exotic’ ones you may want to look at:

Classical Mechanics, 3rd edition by H. Goldstein, C. Poole, and J. Safko. This has long
been the standard for graduate classical mechanics and the new edition also covers aspects
of nonlinear dynamics.

Theoretical Mechanics of Particles and Continua, Alexander L. Fetter and John D.
Walecka. The approach here is one of emphasizing the transition from discrete to
continuum mechanics and the newest version comes with a supplement on nonlinear
dynamics.

Mechanics, L.D. Landau and E. M. Lifshitz. Wonderful text brimming with insight but may
be better to use as a reference text rather than a primary textbook.

Chaos and Integrability in Nonlinear Dynamics: An Introduction, Michael J. Tabor. Very
well-written and a good way to be introduced to the modern perspective on classical
mechanics.

Classical Dynamics: a contemporary approach, J. José and E.J. Saletan. Focus is on
nonlinear dynamics and the book places emphasis on a geometrical approach.

Introduction to Dynamics, I. C. Percival & D. Richards. Another text book with a modern
take but at a lower level.
The course starts from Newtonian concepts encountered in the introductory physics classes and
ends with the outline of a more general framework which plays an important role in several
branches of physics (i.e statistical physics, quantum mechanics, field theories etc.). Along the way,
we will also encounter ideas key to many engineering sub-fields as well. At the end, the notion that
the field is not fossilized but is an area of active research (nonlinear dynamics) will be introduced.
Grading policy: There will be 8 homework assignments which will be handed out with (strictly
enforced) due dates. Collaboration and discussion while solving these problems is encouraged with
the usual caveat that the final work turned in be independently completed. Performance on these
assignments will constitute 40% of the final grade. A mid-term exam (20%) near the 8th week of
the semester and a comprehensive final exam (40%) will make up the remainder of the grade.
Detailed Syllabus:
Week 1:
o Preliminary remarks and review of concepts encountered earlier
o extensions to systems of particles
o role and inclusion of constraints; D’Alembert’s principle
Week 2:
o Lagrangian methods: canonical examples, e.g. bead on a hoop
o Calculus of variations; Hamilton’s principle with constraints
Week 3:
o Examples with one or more constraints
o Noether’s Theorem and conservation laws
o Central force problem; Closed orbits; Virial theorem
Week 4:
o Bertrand’s theorem
o Scattering in a central force field
Week 5:
o Coordinate transformations; Accelerated frames; Corioli’s force
o Small oscillations and normal mode analysis
Week 6:
o Small oscillations and normal modes continued
o Examples
o Discussion of Jacobi matrices
Week 7:
o discussion of discrete to continuum extension
o Hamilton’s principle for continuous systems
o Forced vibrations and dissipation
o Dynamics of differential equations; integration by quadrature
o Introduction to Elliptic functions
Week 8:
o Phase space methods and linear stability analysis
o MID-TERM EXAM
Week 9:
o Fixed points and their classification
o Hamiltonian dynamics; Legendre transformations
o Principle of least action; Canonical transformations; examples; Poisson brackets
Week 10:
o Generating function and optimal transformations
o Hamilton Jacobi theory and applications
o Action-Angle variables
Week 11:
o Integrability and separability
o Properties of Integrable systems
o Toroidal motion
o Poincaré-Hopf theorem
o Canonical Perturbation Theory
Week 12:
o Digression on perturbation series, regular and singular
o Perturbation series for the Hamilton-Jacobi equation
o Perturbed Harmonic oscillator
o Problem of small divisors
Week 13:
o Kolmogorov-Arnold-Moser (KAM) theorem
o Superconvergent perturbation theory
o Related number-theoretic considerations
Week 14:
o Chaos: the final frontier
o Chaos in Hamiltonian systems and Area-preserving mappings
o Chirikov-Taylor or standard mapping
o Poincaré-Birkhoff theorem; homoclinic points
o Criteria for local chaos: Lyapunov exponents; KS entropy
o Local to Global chaos?
Accommodations
The University of Massachusetts Boston is committed to providing reasonable academic
accommodations for all students with disabilities. This syllabus is available in alternate format upon
request. If you have a disability and feel you will need accommodations in this course, please contact the
Ross Center for Disability Services, Campus Center, Upper Level, Room 211 at 617.287.7430.
http://www.umb.edu/academics/vpass/disability/ After registration with the Ross Center, a student
should present and discuss the accommodations with the professor. Although a student can request
accommodations at any time, we recommend that students inform the professor of the need for
accommodations by the end of the Drop/Add period to ensure that accommodations are available for the
entirety of the course.
Code of Conduct and Academic Integrity
It is the expressed policy of the University that every aspect of academic life--not only formal
coursework situations, but all relationships and interactions connected to the educational process--shall
be conducted in an absolutely and uncompromisingly honest manner. The University presupposes that
any submission of work for academic credit is the student’s own and is in compliance with University
policies, including its policies on appropriate citation and plagiarism. These policies are spelled out in
the Code of Student Conduct. Students are required to adhere to the Code of Student Conduct, including
requirements for academic honesty, as delineated in the University of Massachusetts Boston Graduate
Catalogue and relevant program student handbook(s)
http://www.umb.edu/life_on_campus/policies/code.