Textbooks: L. Landau and E. Lifshitz, Course of Theoretical Physics, Vol.1, Mechanics; D. Morin, Classical Mechanics; H. Goldstein, Classical Mechanics; S. Strogatz, Nonlinear Dynamics and Chaos.
[13 Apr 2021 onwards, thro late July]
Rough outline of content based on previous semesters (actual content may evolve and change as the lecs proceed):
The Lagrangian formulation of classical mechanics; principle of least action and the Euler-Lagrange equation etc, symmetries and conservation laws; 2-body problem and central forces.
Special relativity -- broad context, basic effects (time dilation, length contraction, loss of simultaneity), Lorentz transformations and the invariance of the interval, spacetime diagrams and lightcones, relativistic particle Lagrangian and rudiments of relativistic dynamics. (quick glimpse: deriving Lorentz force law equation from a charged particle Lagrangian)
Small oscillations;
Rotating frames and the Coriolis force; Rotations and rigid bodies: rotational kinetic energy, angular momentum and the inertia tensor; Euler angle parametrization; free rotation of a symmetric top, precession etc; dynamics and Euler's equations.
Hamiltonian formulation of classical dynamics, Hamilton equations, Poisson brackets; the action as a function of final coordinates/time; canonical transformations; infinitesimal canonical transformations and their generators (e.g. Hamiltonian as generator of time translations).
Dynamical systems: this module is based on Strogatz's book (sections
listed below).
Hamilton equations and phase space flowlines (e.g. harmonic oscillator); 1-D maps, graphical methods, flowlines, fixed points and stability [sec.2.1,2.2,2.4]; 2-d flows: fixed points, stability and classification of linear flows (specifically saddle points and stable/unstable nodes, and relation to eigenvalues/eigenvectors) [parts of ch.5];
Nonlinear flows and phase portraits: simple examples, fixed points and linearization in their neighbourhood, extrapolation to full phase portrait, validity thereof [sec.6.1,6.2,6.3]; Simple biological systems: population growth and the logistic equation [sec.2.3]; toy example of Lotka-Volterra model of competition between two species (rabbits vs sheep) and phase portrait [sec.6.4];
Discrete/iterative maps, fixed points, stability, cobweb diagrams. Logistic map [x(n+1)=r.xn.(1-xn)], fixed points, bifurcations, period doubling, chaos [sec.10.1-10.4]. Numerical plots on Mathematica and evolution for various values of r, onset of chaotic behaviour, periodic window within etc.
Assignments: 35%
Midsem exam, 30%,
Endsem exam, 35%.