University of Oxford Physics Year 3 - Hamiltonian Mechanics (B7) Complete Lecture Notes & Revision Guide

Comprehensive handwritten lecture notes for the University of Oxford Year 3 Physics course "Classical Mechanics (B7)". These notes provide a structured and detailed treatment of the material covered throughout the course, including derivations, worked examples, key formulas, diagrams, and explanations designed to support both understanding and revision. Topics covered include: • Calculus of variations, functionals, variational principles, Euler-Lagrange equations, Beltrami identities, and constrained variational problems using Lagrange multipliers • Classical variational problems including shortest paths, brachistochrone curves, hanging chains, constrained systems, and the physical interpretation of variational methods • Hamilton's principle of least action, action functionals, generalized coordinates, generalized velocities, configuration space, and the foundations of analytical mechanics • Lagrangian mechanics, derivation of equations of motion, generalized momenta, generalized forces, cyclic coordinates, and conservation laws arising from symmetries • Applications of Lagrangian methods to particles in non-Cartesian coordinate systems including spherical coordinates, rotating reference frames, and systems with constraints • Coriolis forces, centrifugal forces, rotating coordinate systems, and the treatment of non-inertial motion within the Lagrangian framework • Rigid body dynamics, centre-of-mass motion, inertia tensors, principal axes, angular momentum, rotational kinetic energy, and Euler angles • Euler's equations for rigid-body rotation, stability of rotating bodies, tennis-racket theorem, symmetric tops, precession, and rotational motion of rigid systems • Coupled oscillators, normal modes, normal frequencies, mode decomposition, and stability analysis of multi-degree-of-freedom systems • Noether's theorem, continuous symmetries, conserved quantities, translational symmetry, rotational symmetry, temporal symmetry, and the deep connection between symmetry and conservation laws • Hamiltonian mechanics, Legendre transforms, canonical coordinates, Hamiltonians, Hamilton's equations, and the reformulation of dynamics in phase space • Phase-space dynamics, Liouville's theorem, incompressible phase-space flow, canonical evolution, and conservation of phase-space volume • Poisson brackets, canonical structure, symmetries in Hamiltonian systems, conservation laws, canonical transformations, Hamilton-Jacobi theory, action-angle variables, ray tracing theory, Hamiltonian descriptions of wave propagation, and connections between Hamiltonian mechanics and the derivation of Schrödinger's equation The document consists of carefully organised handwritten notes taken during the course and is suitable for lecture review, tutorial preparation, revision, and exam preparation. Ideal for Oxford Physics students and anyone studying advanced classical mechanics, analytical mechanics, Hamiltonian dynamics, symmetry methods, and the mathematical foundations of theoretical physics.