\contentsline {chapter}{\numberline {1}From Newton's Laws to Langrange's Equations}{2}{chapter.1}
\contentsline {section}{\numberline {1.1}Lagrangian and Newtonian Approaches}{2}{section.1.1}
\contentsline {subsection}{\numberline {1.1.1}Lagrangian versus Hamiltonian Approaches}{6}{subsection.1.1.1}
\contentsline {section}{\numberline {1.2}Prehistory of the Lagrangian Approach}{6}{section.1.2}
\contentsline {subsection}{\numberline {1.2.1}The Principle of Minimum Energy}{7}{subsection.1.2.1}
\contentsline {subsection}{\numberline {1.2.2}D'Alembert's Principle and Lagrange's Equations}{8}{subsection.1.2.2}
\contentsline {subsection}{\numberline {1.2.3}The Principle of Least Time}{10}{subsection.1.2.3}
\contentsline {subsection}{\numberline {1.2.4}How D'Alembert and Others Got to the Truth}{12}{subsection.1.2.4}
\contentsline {chapter}{\numberline {2}Equations of Motion}{14}{chapter.2}
\contentsline {section}{\numberline {2.1}The Euler-Lagrange Equations}{14}{section.2.1}
\contentsline {subsection}{\numberline {2.1.1}Comments}{16}{subsection.2.1.1}
\contentsline {subsection}{\numberline {2.1.2}Lagrangian Dynamics}{16}{subsection.2.1.2}
\contentsline {section}{\numberline {2.2}Interpretation of Terms}{18}{section.2.2}
\contentsline {chapter}{\numberline {3}Lagrangians and Noether's Theorem}{20}{chapter.3}
\contentsline {section}{\numberline {3.1}Time Translation}{20}{section.3.1}
\contentsline {subsection}{\numberline {3.1.1}Canonical and Generalized Coordinates}{21}{subsection.3.1.1}
\contentsline {subsubsection}{Generalized Coordinates}{21}{section*.2}
\contentsline {subsubsection}{Canonical Coordinates}{22}{section*.3}
\contentsline {section}{\numberline {3.2}Symmetry and Noether's Theorem}{22}{section.3.2}
\contentsline {paragraph}{Remark:}{23}{section*.4}
\contentsline {subsection}{\numberline {3.2.1}Noether's Theorem}{23}{subsection.3.2.1}
\contentsline {subsubsection}{Example}{24}{section*.5}
\contentsline {section}{\numberline {3.3}Conserved Quantities from Symmetries}{24}{section.3.3}
\contentsline {subsection}{\numberline {3.3.1}Time Translation Symmetry}{25}{subsection.3.3.1}
\contentsline {subsection}{\numberline {3.3.2}Space Translation Symmetry}{25}{subsection.3.3.2}
\contentsline {paragraph}{Aside:}{26}{section*.6}
\contentsline {subsection}{\numberline {3.3.3}Rotational Symmetry}{26}{subsection.3.3.3}
\contentsline {section}{\numberline {3.4}Example Problems}{28}{section.3.4}
\contentsline {subsection}{\numberline {3.4.1}The Atwood Machine}{28}{subsection.3.4.1}
\contentsline {subsection}{\numberline {3.4.2}Disk Pulled by Falling Mass}{29}{subsection.3.4.2}
\contentsline {subsection}{\numberline {3.4.3}Free Particle in Special Relativity}{31}{subsection.3.4.3}
\contentsline {subsubsection}{Comments}{33}{section*.7}
\contentsline {section}{\numberline {3.5}Electrodynamics and Relativistic Lagrangians}{35}{section.3.5}
\contentsline {subsection}{\numberline {3.5.1}Gauge Symmetry and Relativistic Hamiltonian}{35}{subsection.3.5.1}
\contentsline {subsection}{\numberline {3.5.2}Relativistic Hamiltonian}{36}{subsection.3.5.2}
\contentsline {section}{\numberline {3.6}Relativistic Particle in an Electromagnetic Field}{37}{section.3.6}
\contentsline {section}{\numberline {3.7}Alternative Lagrangians}{39}{section.3.7}
\contentsline {subsection}{\numberline {3.7.1}Lagrangian for a String}{39}{subsection.3.7.1}
\contentsline {subsection}{\numberline {3.7.2}Alternate Lagrangian for Relativistic Electrodynamics}{41}{subsection.3.7.2}
\contentsline {paragraph}{Comments.}{42}{section*.8}
\contentsline {section}{\numberline {3.8}The General Relativistic Particle}{43}{section.3.8}
\contentsline {subsection}{\numberline {3.8.1}Free Particle Lagrangian in GR}{44}{subsection.3.8.1}
\contentsline {subsection}{\numberline {3.8.2}Charged particle in EM Field in GR}{45}{subsection.3.8.2}
\contentsline {section}{\numberline {3.9}The Principle of Least Action and Geodesics}{46}{section.3.9}
\contentsline {subsection}{\numberline {3.9.1}Jacobi and Least Time vs Least Action}{46}{subsection.3.9.1}
\contentsline {subsection}{\numberline {3.9.2}The Ubiquity of Geodesic Motion}{48}{subsection.3.9.2}
\contentsline {chapter}{\numberline {4}From Lagrangians to Hamiltonians}{51}{chapter.4}
\contentsline {section}{\numberline {4.1}The Hamiltonian Approach}{51}{section.4.1}
\contentsline {section}{\numberline {4.2}Regular and Strongly Regular Lagrangians}{54}{section.4.2}
\contentsline {subsection}{\numberline {4.2.1}Example: A Particle in a Riemannian Manifold with Potential $V(q)$}{54}{subsection.4.2.1}
\contentsline {subsection}{\numberline {4.2.2}Example: General Relativistic Particle in an E-M Potential}{55}{subsection.4.2.2}
\contentsline {subsection}{\numberline {4.2.3}Example: Free General Relativistic Particle with Reparameterization Invariance}{55}{subsection.4.2.3}
\contentsline {subsection}{\numberline {4.2.4}Example: A Regular but not Strongly Regular Lagrangian}{55}{subsection.4.2.4}
\contentsline {section}{\numberline {4.3}Hamilton's Equations}{56}{section.4.3}
\contentsline {subsection}{\numberline {4.3.1}Hamilton and Euler-Lagrange}{57}{subsection.4.3.1}
\contentsline {subsubsection}{Example: Particle in a Potential $V(q)$}{58}{section*.9}
\contentsline {subsubsection}{Note on Symplectic Structure}{59}{section*.10}
\contentsline {subsection}{\numberline {4.3.2}Hamilton's Equations from the Principle of Least Action}{59}{subsection.4.3.2}
\contentsline {section}{\numberline {4.4}Waves versus Particles---The Hamilton-Jacobi Equations}{61}{section.4.4}
\contentsline {subsection}{\numberline {4.4.1}Wave Equations}{61}{subsection.4.4.1}
\contentsline {subsection}{\numberline {4.4.2}The Hamilton-Jacobi Equations}{63}{subsection.4.4.2}