Classical Mechanics II

Classical Mechanics II (Spring '19)

Textbooks: L. Landau and E. Lifshitz, Course of Theoretical Physics, Vol.1, Mechanics;   H. Goldstein, Classical Mechanics;   D. Morin, Classical Mechanics;   L. Hand and J. Finch, Analytical Mechanics;   S. Strogatz, Nonlinear Dynamics and Chaos.

Rough outline of content:

Review of the Lagrangian formulation;

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. Deriving Lorentz force law equation from a charged particle Lagrangian.

Review of central forces, small oscillations;

Hamiltonian formulation of classical dynamics, Hamilton equations; 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).

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; heavy symmetric top from a Lagrangian point of view, effective potential and precession, nutation etc (from Hand/Finch);

Dynamical systems: this module is based on Strogatz's book (sections listed below).

1-D maps, graphical methods, flowlines, fixed points and stability [sec.2.1,2.2,2.4];

bifurcations (saddle nodes, transcritical, pitchfork etc) [sec.3.1,3.2,3.4 and glimpse of 3.6];

2-d flows: fixed points, stability and classification of linear flows (saddle points, nodes, spirals, and relation to eigenvalues/eigenvectors) [all 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]; simple model for the evolution of an epidemic (ex.3.7.6); toy example of Lotka-Volterra model of competition between two species (rabbits vs sheep) and phase portrait [sec.6.4];

2-d flows and bifurcations (saddle-node, supercritical pitchfork etc) [sec.8.1];

Limit cycles --- simple explicit example and van der Pol oscillator [sec.7.1]; Ruling out closed orbits and gradient flows, Lipapunov functions sec.7.2]; supercritical Hopf bifurcation [sec.8.2];

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%.

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