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1. MATHEMATICAL METHODS FOR PHYSICS
Unit I
Complex Variables:
Complex numbers. Equations to curves in the plane in terms of z and z*. The Riemann sphere
and stereographic projection. Analytic functions of z and the Cauchy Riemann conditions. The
real and imaginary parts of an analytic function. The derivative of an analytic function. Power
series as analytic functions. Convergence of power series. Cauchy's integral theorem.
Singularities, removable singularity, simple pole, multiple pole, essential singularity. Laurent
series. Singularity at infinity. Accumulation point of poles. Meromorphic functions. Cauchy's
integral formula Solution of differential equations using generating functions and contour
integration. Summation of series using contour integration. Evaluation of definite integrals using
contour integration.
Unit II
Gamma function
Definition of the gamma function and its analytic continuation. Analytic properties. Connection
with gaussian integrals. Mittag-Leffler expansion of the gamma function. Logarithmic
derivative of the gamma function. Infinite product representation for gamma function. beta
function. Reflection and duplication formulas for gamma function.
Laplace transforms and Green Function
Definition of the Laplace transform. The convolution theorem. Laplace transforms of derivatives.
The inverse transform, Mellin's formula. The LCR series circuit. Laplace transform of the Bessel
and modified Bessel functions of the first kind. Laplace transforms and random processes: the
Poisson process. Laplace transforms and random processes: biased random walk on a linear
lattice and on a d-dimensional lattice.
Green functions. Poisson's equation. The fundamental Green function for thelaplacian operator.
Solution of Poisson's equation for a spherically symmetric source. The Coulomb potential in d
dimensions. Ultraspherical coordinates. A divergence problem. Dimensional regularization.
Direct derivation using Gauss' Theorem. The Coulomb potential in d = 2 dimensions.
Text Books
1. L.A. Pipes and L.R. Harvill, Applied Mathematics for Engineers and Physicists, McGraw-Hill (1970).
2. G. B. Arfken and H.J. Weber, Mathematical Methods for Physicists, 5th edition, Academic Press (2001).
3. E. Kreyszig, Advanced Engineering Mathematics, 8th edition, John Wiley & Sons Inc. (1999).
4. W.W Bell: Special functions for scientists and engineers.
2. CLASSICAL MECHANICS
Unit I
Lagrangian Formulation
Newtonian mechanics and its limitations. Constrained motion. Constraints and their
classification. Principle of virtual work. D’ Alembert’s principle. Generalized coordinates.
Deduction of Lagrange’s equations from D’ Alembert’s Principle. Generalized momenta and
energy. Cyclic or ignorable coordinates. Rayleigh’ s dissipation function. Integrals of motion.
Symmetries of space and time with conservation laws. Problems. Rotating frames. Inertial
Forces. Electromagnetic analogy of inertial forces. Terrestrial and astronomical applications of
Conolis force. Foucault’s pendulum. Problems.
Unit II
Central Force Problem
Central force. Definition and properties of central force. Two-body central force problem.
Stability of orbits. Conditions for closure. General analysis of orbits. Kepler’s laws. Kepler’s
equation. Artificial satellites. Rutherford scattering. Problems.
Principle of least action. Hamilton’s principle. The calculas of variations. Derivation of
Hamilton’s equations of motion for holonomic systems from hamilton’s principle. Hamilton’s
principle and characteristic functions.
UNIT III
Canonical Transformations
Generating functions. Poisson bracket. Poisson’s Theorem. Invariance of PB under canonical
transformations. Angular momentum PBs. Hamilton-Jacobi equation. Connection with Classical
Mechanics canonical transformation. Problems. Small oscillations. Normal modes and
coordinates. Problems.
Principles and postulates of relativity, Lorentz Transformation, Effects thereof, Tensors,
transformation properties, symmetric and anti-symmetric properties, Four Vector notation,
Energy Momentum four vector for a particle, relativistic invariance of Physical Laws.
Lagrangian and Hamiltonian of a relativistic particle.
Text Books
1. H. Goldstein, C. poole and J. Safko, Classical Mechanics, 3nd edition, Addison & Wesley(2000).
2. L.D. Landau and E.M. Lifshitz, Mechanics, Buttorworth-Heinemann (1976).
3. W. Greiner, Classical Mechanics – Point particles and Relativity, Springer-Verlag (1989).
4. N.C Rana and P.S Joag, Classical Mechanics.
5. A.P French: Special Relativity.
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