The Principles of Dynamics

Introduction: Sin and Death From Newton to Lagrange Summary of Newton's Laws Momentum, Work and Forces Systems of Particles Energy Constraints D'Alembert's Principle and Lagrange's Equations Two preliminary lemmas D'Alembert's Principle Lagrange's Equations Generalisations of Lagrange's Equations Velocity Dependent Potentials Constraints Lagrange's Equations for Impulsive Forces Some Definitions Hamilton's Principle Hamilton's Principle Conservation Laws and Symmetries Rigid Bodies Frame of Reference Rotating Frames Transforming Between Frames Euler's Theorem The Moment of Inertia Properties of the Moment of Inertia Tensor Spinning Tops Deriving the Lagrangian Conserved Quantities Steady Motion Stability Investigation Hamilton's Equations, & Onwards to Abstraction Hamilton's Equations An alternative approach Cyclic Coordinates and Conservation Theorems The principle of Least Action Canonical Transformations Canonical Transformations Generalisation to Higher Dimensions Poisson Brackets The Sympletic Condition and CTs The Special Case The General Case More on ICTs The Hamiltonian as the generator of an ICT Symmetry and Conserved Quantities The Hamilton-Jacobi Equation `Nice' coordinates The principle Function and the Lagrangian Integrable Systems Integrable Systems Action-Angle Variables Adiabatic Invariants