\contentsline {section}{\tocsection {}{1}{Oct 3, 2006: Introduction}}{3}{section.1}
\contentsline {subsection}{\tocsubsection {}{1.1}{Web information}}{3}{subsection.1.1}
\contentsline {subsection}{\tocsubsection {}{1.2}{Perspective}}{3}{subsection.1.2}
\contentsline {subsection}{\tocsubsection {}{1.3}{(Higher) cohomology and physics}}{4}{subsection.1.3}
\contentsline {subsection}{\tocsubsection {}{1.4}{Classical dynamics vs.\nonbreakingspace open string statics}}{5}{subsection.1.4}
\contentsline {subsection}{\tocsubsection {}{1.5}{The quantum case}}{6}{subsection.1.5}
\vspace {2ex}
\contentsline {section}{\tocsection {}{2}{Oct 10, 2006: Lagrangian Mechanics}}{7}{section.2}
\contentsline {subsection}{\tocsubsection {}{2.1}{Introduction to the Lagrangian approach}}{7}{subsection.2.1}
\contentsline {subsection}{\tocsubsection {}{2.2}{Deriving the Euler-Lagrange equations}}{8}{subsection.2.2}
\contentsline {subsection}{\tocsubsection {}{2.3}{Physics notation}}{9}{subsection.2.3}
\contentsline {subsection}{\tocsubsection {}{2.4}{Example: A particle in a potential}}{9}{subsection.2.4}
\contentsline {subsection}{\tocsubsection {}{2.5}{``Sneak preview"}}{10}{subsection.2.5}
\vspace {2ex}
\contentsline {section}{\tocsection {}{3}{Oct 17, 2006: From Lagrangian to Hamiltonian Dynamics}}{11}{section.3}
\contentsline {subsection}{\tocsubsection {}{3.1}{Recap}}{11}{subsection.3.1}
\contentsline {subsection}{\tocsubsection {}{3.2}{A matter of notation}}{11}{subsection.3.2}
\contentsline {subsection}{\tocsubsection {}{3.3}{Switching to the Hamiltonian approach}}{12}{subsection.3.3}
\contentsline {subsection}{\tocsubsection {}{3.4}{Energy}}{12}{subsection.3.4}
\contentsline {subsection}{\tocsubsection {}{3.5}{Hamilton's Equations}}{13}{subsection.3.5}
\vspace {2ex}
\contentsline {section}{\tocsection {}{4}{Oct 24, 2006: Hamiltonian Mechanics and Symplectic Geometry}}{14}{section.4}
\contentsline {subsection}{\tocsubsection {}{4.1}{Recap}}{14}{subsection.4.1}
\contentsline {subsection}{\tocsubsection {}{4.2}{Some musical operators}}{14}{subsection.4.2}
\contentsline {subsection}{\tocsubsection {}{4.3}{The Hamiltonian vector field}}{15}{subsection.4.3}
\contentsline {subsection}{\tocsubsection {}{4.4}{Homework}}{16}{subsection.4.4}
\contentsline {subsection}{\tocsubsection {}{4.5}{Coordinate-free formulations}}{17}{subsection.4.5}
\vspace {2ex}
\contentsline {section}{\tocsection {}{5}{Oct 31, 2006: More on the canonical 1-form}}{18}{section.5}
\contentsline {subsection}{\tocsubsection {}{5.1}{Reconciling with the coordinate-based definition}}{18}{subsection.5.1}
\contentsline {subsection}{\tocsubsection {}{5.2}{Symplectic manifolds}}{19}{subsection.5.2}
\contentsline {subsection}{\tocsubsection {}{5.3}{Digression on five-body systems}}{20}{subsection.5.3}
\contentsline {subsection}{\tocsubsection {}{5.4}{The 1-form and action}}{21}{subsection.5.4}
\vspace {2ex}
\contentsline {section}{\tocsection {}{6}{Nov 07, 2006: The Extended Phase Space}}{22}{section.6}
\contentsline {subsection}{\tocsubsection {}{6.1}{Aside}}{22}{subsection.6.1}
\contentsline {subsection}{\tocsubsection {}{6.2}{Bringing in spacetime}}{22}{subsection.6.2}
\contentsline {subsection}{\tocsubsection {}{6.3}{Hamilton's equations and the conservation of energy}}{23}{subsection.6.3}
\contentsline {subsection}{\tocsubsection {}{6.4}{Digression of the day: LIGO}}{25}{subsection.6.4}
\contentsline {subsection}{\tocsubsection {}{6.5}{A look back at the special case $t(s) = s$}}{25}{subsection.6.5}
\vspace {2ex}
\contentsline {section}{\tocsection {}{7}{Nov 14, 2006: From particles to strings and higher membranes}}{27}{section.7}
\contentsline {subsection}{\tocsubsection {}{7.1}{More derivations}}{28}{subsection.7.1}
\contentsline {subsection}{\tocsubsection {}{7.2}{Generalizing the Lagrangian}}{29}{subsection.7.2}
\vspace {2ex}
\contentsline {section}{\tocsection {}{8}{Nov 28, 2006: More on particles ``vs." membranes}}{30}{section.8}
\contentsline {subsection}{\tocsubsection {}{8.1}{(Functorial) construction of the multivelocity alternating tensor}}{30}{subsection.8.1}
\contentsline {subsection}{\tocsubsection {}{8.2}{Volume forms}}{30}{subsection.8.2}
\contentsline {subsection}{\tocsubsection {}{8.3}{The canonical $p$-form}}{33}{subsection.8.3}
\vspace {2ex}
\contentsline {section}{\tocsection {}{9}{Dec 05, 2006: Phases and connections on bundles}}{36}{section.9}
\contentsline {subsection}{\tocsubsection {}{9.1}{When connections come in}}{36}{subsection.9.1}
\contentsline {subsection}{\tocsubsection {}{9.2}{Phases and relative phases}}{36}{subsection.9.2}
\contentsline {subsection}{\tocsubsection {}{9.3}{Example: Rigid rotor}}{38}{subsection.9.3}
\contentsline {subsection}{\tocsubsection {}{9.4}{Integral cohomology and Max Planck}}{39}{subsection.9.4}