Classical Mechanics: Lagrangian & Hamiltonian Formulations

Learning Objectives

• Formulate mechanical problems using generalised coordinates and D'Alembert's principle
• Derive and apply the Euler-Lagrange equations from Hamilton's variational principle
• Identify cyclic coordinates and apply Noether's theorem to find conserved quantities
• Analyse small oscillations and normal modes of coupled mechanical systems
• Construct the Hamiltonian via the Legendre transform and apply canonical equations
• Perform canonical transformations using generating functions and the symplectic condition
• Apply the Hamilton-Jacobi equation to integrable systems and derive action-angle variables
• Use Poisson brackets to characterise constants of motion and the symplectic structure
• Analyse rigid body rotation using the inertia tensor, Euler angles, and Euler's equations
• Connect classical mechanics to quantum mechanics through canonical quantisation and WKB theory