SYLLABUS (tentative)
Vectors and vector fields, products of vectors, line integral of a vector field, concept of scalar potential, surface and volume integrals of vector fields, divergence and Stoke's theorems, Green’s identities, particle kinematics, rotation of coordinate axes, invariants under rotations, curvilinear coordinates: plane polar, cylindrical and spherical polar coordinates, vector derivatives: velocity and acceleration vectors expressed in terms of polar coordinates, concept of angular velocity
Newtonian mechanics: some examples of particle dynamics, retarding forces, motion under frictional drag force, periodic motion, harmonic and anharmonic oscillators, conservative property of an oscillator, damped oscillator, under and overdamped motions, effect of a harmonic driving force on a damped oscillator, resonant behavior
Coupled oscillators with two degrees of freedom, normal modes of oscillation and the relevant characteristic frequencies, the role the boundary conditions play on the characteristic modes of oscillation, two dimensional oscillator with linked coordinates, phenomenon of beats
Classical wave equation, transverse vibrations on strings and membranes, variational approximations to estimate the eigenfrequencies
Wave equation pertaining to oscillations in nonhomogeneous media; examples: lateral vibrations of a hanging chain, transverse vibrations along a rope spun about one end; Bessel functions or Legendre polynomials as characteristic descriptions of vibration,
Oscillatory systems with many degrees of freedom,
Continuous systems, sinusoidal and exponential waves
Gravitation and gravitational potential, Central force motion
Dynamics of a system of particles, the centre of mass concept, elastic and inelastic collisions
Calculus of variations, Euler's equations, the Brachistochrone problem
Hamilton's principle, Lagrange's equations of motion in generalised coordinates, Hamiltonian's equations of motion, Lagrangian and Hamiltonian dynamics: examples
Rigid body dynamics, inertia tensor, principle axes of inertia, Euler's equations of motion for a rigid body
