MA22S4: Mechanics (Hilary Term 2018)

Scalar and vector products, differentiation and integration of vectors, velocity and acceleration, Newton's Laws.

Derivation of velocity and acceleration in polar coordinates and applications to circular and elliptical motion of a particle.

Equations of motion for a particle in a central force field, derivation of the orbit equation, conservation of angular momentum, potential energy, conservation of energy, solution of the orbit equation for different force fields, apsides and apsidal angles, calculation of maximum and minimum distance of a particle from the origin of a force, inverse square law of attraction and conic sections, properties of the ellipse. Planetary motion, Newton's Universal Law of Gravitation, proof of Kepler's Laws, examples involving calculating eccentricity, periodic time, velocity at aphelion and perihelion of planets and related problems.

Evaluation of work done by a force on a particle using line integrals, work as related to kinetic and potential energy, conservative forces, path independence, conservation of energy. Energy diagrams -- use of energy diagrams to analyse the motion of a particle qualitatively, positions of stable and unstable equilibrium, small oscillations in a bound system.

Non-inertial coordinate systems, velocity and acceleration in rotating systems, centrifugal and coriolis forces, derivation of the equation of motion for a particle moving in the vicinity of the rotating earth and related examples.

Generalised coordinates, Lagrangian and the action of the system, Hamilton's principle, derivation of Euler-Lagrange equation and some applications. Generalised forces and generalised momenta. Hamiltonian and Hamilton's equations.