\contentsline {chapter}{\numberline {1}From Newtonian to Lagrangian Mechanics}{1}{chapter.1}
\contentsline {section}{\numberline {1.1}Lagrangian and Newtonian Approaches}{1}{section.1.1}
\contentsline {subsection}{\numberline {1.1.1}Lagrangian versus Hamiltonian Approaches}{5}{subsection.1.1.1}
\contentsline {section}{\numberline {1.2}Prehistory of the Lagrangian Approach}{5}{section.1.2}
\contentsline {subsection}{\numberline {1.2.1}The Principle of Least Time}{7}{subsection.1.2.1}
\contentsline {subsection}{\numberline {1.2.2}The Principle of Minimum Energy}{9}{subsection.1.2.2}
\contentsline {subsection}{\numberline {1.2.3}Virtual Work}{10}{subsection.1.2.3}
\contentsline {subsection}{\numberline {1.2.4}From Virtual Work to the Principle of Least Action}{11}{subsection.1.2.4}
\contentsline {chapter}{\numberline {2}Lagrangian Mechanics}{15}{chapter.2}
\contentsline {section}{\numberline {2.1}The Euler--Lagrange Equations}{15}{section.2.1}
\contentsline {section}{\numberline {2.2}Noether's Theorem}{20}{section.2.2}
\contentsline {subsection}{\numberline {2.2.1}Time Translation}{20}{subsection.2.2.1}
\contentsline {subsubsection}{Generalized Coordinates}{21}{section*.3}
\contentsline {subsection}{\numberline {2.2.2}Symmetries}{21}{subsection.2.2.2}
\contentsline {paragraph}{Remark:}{22}{section*.4}
\contentsline {subsection}{\numberline {2.2.3}Noether's Theorem}{22}{subsection.2.2.3}
\contentsline {subsection}{\numberline {2.2.4}Conservation of Energy}{22}{subsection.2.2.4}
\contentsline {section}{\numberline {2.3}Conserved Quantities from Symmetries}{23}{section.2.3}
\contentsline {subsection}{\numberline {2.3.1}Time Translation Symmetry}{24}{subsection.2.3.1}
\contentsline {subsection}{\numberline {2.3.2}Space Translation Symmetry}{24}{subsection.2.3.2}
\contentsline {paragraph}{Aside:}{25}{section*.6}
\contentsline {subsection}{\numberline {2.3.3}Rotational Symmetry}{25}{subsection.2.3.3}
\contentsline {chapter}{\numberline {3}Examples}{27}{chapter.3}
\contentsline {section}{\numberline {3.1}The Atwood Machine}{27}{section.3.1}
\contentsline {section}{\numberline {3.2}Bead on a Rotating Rod}{28}{section.3.2}
\contentsline {section}{\numberline {3.3}Disk Pulled by Falling Mass}{30}{section.3.3}
\contentsline {section}{\numberline {3.4}Free Particle in Special Relativity}{32}{section.3.4}
\contentsline {subsection}{\numberline {3.4.1}Comments}{34}{subsection.3.4.1}
\contentsline {section}{\numberline {3.5}Gauge Symmetries}{36}{section.3.5}
\contentsline {subsection}{\numberline {3.5.1}Relativistic Hamiltonian}{37}{subsection.3.5.1}
\contentsline {section}{\numberline {3.6}Relativistic Particle in an Electromagnetic Field}{38}{section.3.6}
\contentsline {section}{\numberline {3.7}Lagrangian for a String}{40}{section.3.7}
\contentsline {section}{\numberline {3.8}Another Lagrangian for Relativistic Electrodynamics}{41}{section.3.8}
\contentsline {paragraph}{Comments.}{43}{section*.7}
\contentsline {section}{\numberline {3.9}The Free Particle in General Relativity}{44}{section.3.9}
\contentsline {section}{\numberline {3.10}A Charged Particle on a Curved Spacetime}{46}{section.3.10}
\contentsline {section}{\numberline {3.11}The Principle of Least Action and Geodesics}{46}{section.3.11}
\contentsline {subsection}{\numberline {3.11.1}Jacobi and Least Time vs Least Action}{46}{subsection.3.11.1}
\contentsline {subsection}{\numberline {3.11.2}The Ubiquity of Geodesic Motion}{49}{subsection.3.11.2}
\contentsline {chapter}{\numberline {4}From Lagrangians to Hamiltonians}{53}{chapter.4}
\contentsline {section}{\numberline {4.1}The Hamiltonian Approach}{53}{section.4.1}
\contentsline {section}{\numberline {4.2}Regular and Strongly Regular Lagrangians}{56}{section.4.2}
\contentsline {subsection}{\numberline {4.2.1}Example: A Particle in a Riemannian Manifold with Potential $V(q)$}{56}{subsection.4.2.1}
\contentsline {subsection}{\numberline {4.2.2}Example: General Relativistic Particle in an E-M Potential}{56}{subsection.4.2.2}
\contentsline {subsection}{\numberline {4.2.3}Example: Free General Relativistic Particle with Reparameterization Invariance}{57}{subsection.4.2.3}
\contentsline {subsection}{\numberline {4.2.4}Example: A Regular but not Strongly Regular Lagrangian}{57}{subsection.4.2.4}
\contentsline {section}{\numberline {4.3}Hamilton's Equations}{58}{section.4.3}
\contentsline {subsection}{\numberline {4.3.1}Hamilton and Euler--Lagrange}{59}{subsection.4.3.1}
\contentsline {subsubsection}{Example: Particle in a Potential $V(q)$}{60}{section*.8}
\contentsline {subsubsection}{Note on Symplectic Structure}{61}{section*.9}
\contentsline {subsection}{\numberline {4.3.2}Hamilton's Equations from the Principle of Least Action}{61}{subsection.4.3.2}
\contentsline {section}{\numberline {4.4}Waves versus Particles---The Hamilton-Jacobi Equations}{63}{section.4.4}
\contentsline {subsection}{\numberline {4.4.1}Wave Equations}{63}{subsection.4.4.1}
\contentsline {subsection}{\numberline {4.4.2}The Hamilton-Jacobi Equations}{65}{subsection.4.4.2}