Statics Problems: a Tight String, the Catenary, and the Ideal Arch.

Calculus of Variations:  Soap Film, Catenary, Brachistochrone.

Lagrange Multipliers.

Fermat’s Principle of Least Time (wave propagation): Wave Optics and Geometric Optics.

Laws of Dynamics:  Galileo, Newton, Lagrange.

Hamilton’s Principle (least action).

Degrees of Freedom.

Conservation Laws and Noether’s Theorem.

Mechanical Similarity.

Hamilton’s Formalism: Configuration space, State Space, Phase Space.

Hamilton’s Equations, the Canonical Form.

Poisson Brackets, Constants of Motion, the Jacobi Identity.

Adiabatic Invariants.

Action Angle Variables.

Motion in 1D, radial 3D, Kepler.

Bertrand’s Theorem, Precession, Runge Lenz.

Elastic Scattering, Rutherford’s Formula, CM and Lab Frames.

Small Oscillations, Damped, Forced.

Many Variable Small Oscillations: Chain.

Rigid Body Motion, for example sphere on inclined rotating plane.