\BOOKMARK [0][]{chapter.1}{From Newtonian to Lagrangian Mechanics}{} \BOOKMARK [1][]{section.1.1}{Lagrangian and Newtonian Approaches}{chapter.1} \BOOKMARK [2][]{subsection.1.1.1}{Lagrangian versus Hamiltonian Approaches}{section.1.1} \BOOKMARK [1][]{section.1.2}{Prehistory of the Lagrangian Approach}{chapter.1} \BOOKMARK [2][]{subsection.1.2.1}{The Principle of Least Time}{section.1.2} \BOOKMARK [2][]{subsection.1.2.2}{The Principle of Minimum Energy}{section.1.2} \BOOKMARK [2][]{subsection.1.2.3}{Virtual Work}{section.1.2} \BOOKMARK [2][]{subsection.1.2.4}{From Virtual Work to the Principle of Least Action}{section.1.2} \BOOKMARK [0][]{chapter.2}{Lagrangian Mechanics}{} \BOOKMARK [1][]{section.2.1}{The Euler--Lagrange Equations}{chapter.2} \BOOKMARK [1][]{section.2.2}{Noether's Theorem}{chapter.2} \BOOKMARK [2][]{subsection.2.2.1}{Time Translation}{section.2.2} \BOOKMARK [2][]{subsection.2.2.2}{Symmetries}{section.2.2} \BOOKMARK [2][]{subsection.2.2.3}{Noether's Theorem}{section.2.2} \BOOKMARK [2][]{subsection.2.2.4}{Conservation of Energy}{section.2.2} \BOOKMARK [1][]{section.2.3}{Conserved Quantities from Symmetries}{chapter.2} \BOOKMARK [2][]{subsection.2.3.1}{Time Translation Symmetry}{section.2.3} \BOOKMARK [2][]{subsection.2.3.2}{Space Translation Symmetry}{section.2.3} \BOOKMARK [2][]{subsection.2.3.3}{Rotational Symmetry}{section.2.3} \BOOKMARK [0][]{chapter.3}{Examples}{} \BOOKMARK [1][]{section.3.1}{The Atwood Machine}{chapter.3} \BOOKMARK [1][]{section.3.2}{Bead on a Rotating Rod}{chapter.3} \BOOKMARK [1][]{section.3.3}{Disk Pulled by Falling Mass}{chapter.3} \BOOKMARK [1][]{section.3.4}{Free Particle in Special Relativity}{chapter.3} \BOOKMARK [2][]{subsection.3.4.1}{Comments}{section.3.4} \BOOKMARK [1][]{section.3.5}{Gauge Symmetries}{chapter.3} \BOOKMARK [2][]{subsection.3.5.1}{Relativistic Hamiltonian}{section.3.5} \BOOKMARK [1][]{section.3.6}{Relativistic Particle in an Electromagnetic Field}{chapter.3} \BOOKMARK [1][]{section.3.7}{Lagrangian for a String}{chapter.3} \BOOKMARK [1][]{section.3.8}{Another Lagrangian for Relativistic Electrodynamics}{chapter.3} \BOOKMARK [1][]{section.3.9}{The Free Particle in General Relativity}{chapter.3} \BOOKMARK [1][]{section.3.10}{A Charged Particle on a Curved Spacetime}{chapter.3} \BOOKMARK [1][]{section.3.11}{The Principle of Least Action and Geodesics}{chapter.3} \BOOKMARK [2][]{subsection.3.11.1}{Jacobi and Least Time vs Least Action}{section.3.11} \BOOKMARK [2][]{subsection.3.11.2}{The Ubiquity of Geodesic Motion}{section.3.11} \BOOKMARK [0][]{chapter.4}{From Lagrangians to Hamiltonians}{} \BOOKMARK [1][]{section.4.1}{The Hamiltonian Approach}{chapter.4} \BOOKMARK [1][]{section.4.2}{Regular and Strongly Regular Lagrangians}{chapter.4} \BOOKMARK [2][]{subsection.4.2.1}{Example: A Particle in a Riemannian Manifold with Potential V\(q\)}{section.4.2} \BOOKMARK [2][]{subsection.4.2.2}{Example: General Relativistic Particle in an E-M Potential}{section.4.2} \BOOKMARK [2][]{subsection.4.2.3}{Example: Free General Relativistic Particle with Reparameterization Invariance}{section.4.2} \BOOKMARK [2][]{subsection.4.2.4}{Example: A Regular but not Strongly Regular Lagrangian}{section.4.2} \BOOKMARK [1][]{section.4.3}{Hamilton's Equations}{chapter.4} \BOOKMARK [2][]{subsection.4.3.1}{Hamilton and Euler--Lagrange}{section.4.3} \BOOKMARK [2][]{subsection.4.3.2}{Hamilton's Equations from the Principle of Least Action}{section.4.3} \BOOKMARK [1][]{section.4.4}{Waves versus Particles---The Hamilton-Jacobi Equations}{chapter.4} \BOOKMARK [2][]{subsection.4.4.1}{Wave Equations}{section.4.4} \BOOKMARK [2][]{subsection.4.4.2}{The Hamilton-Jacobi Equations}{section.4.4}