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• Use Lagrangian and Hamiltonian formulations to describe continuous systems so as to
understand basic concept of Classical Field Theory.
• Apply theory of relativity to determine time dilation, length contraction and simultaneity.

THEORY (45 Hours)

UNIT I (15 Hours)

Variational Principles and Lagrangian Formulation of Mechanics: D'Alembert's Principle and
Lagrange's equations. Constraints and generalized coordinates. Calculus of variations, Hamilton's
principle and derivation of Lagrange's equation from it. Extension to non-holonomic and non-
conservative systems. Symmetry properties of space and time and the corresponding theorems
(with reference to cyclic coordinates). Simple applications of Lagrangian formulation for a single
particle and a systems of particles. Lagrangian formulation of relativistic mechanics.
Central Force Problem: Equations of motion and first integrals. Equivalent one dimensional
problem andclassification of orbits. The virial theorem. Differential equation for a orbit with a
generalpower law potential. Applications: Kepler problem; scattering in c. m. and lab-coordinates.
UNIT 2 (15 Hours)
Kinematics and Dynamics of Rigid Bodies: Generalized coordinates of a rigid body, orthogonal
transformations and the transformationmatrix. The Euler's angles and Euler's theorem on motion
of rigid bodies, infinitesimalrotations, motion in a rotating frame of reference, Coriolis force on (i)
air flow on thesurface of earth (ii) projectile motion (iii)atomic nuclei. Angular momentum and
Kinetic energy of motion about a point. Moment of inertia tensor, the principle axis
transformation. Euler's equation of motion.
Applications: Torque free motion of a rigid body. Heavy symmetric top with one pointfixed.
Hamilton-Jacobi Theory: The Hamilton-Jacobi equation for (i) Hamilton's principle function, and
(ii) Characteristicsfunction. Separation of variables in Hamilton- Jacobi equation. Action angle
variables.
Applications: Harmonic oscillator with Hamilton-Jacobi and action angle variable methods.
Kepler's problem with action angle variable method.

UNIT 3 (15 Hours)
Hamiltonian Formulation of Mechanics: Legendre's transformations and Hamilton's equations of
motion. Derivation of Hamilton's equations from variational principle. The principle of least
action. Canonical transformations; Poisson's and Lagrangian brackets, their invariance under a
canonical
transformation, equations of motion in the Poisson's bracket notation; infinitesimalcanonical
transformations, constants of motion and symmetry properties.
Applications: Hamiltonian formulation of (i) harmonic oscillator and (ii) relativisticmechanics.
Examples of canonical transformations, with reference to harmonic oscillator. Example of Poisson
bracket, (i) harmonic oscillator; (ii) angular momentum. Lagrangian and Hamiltonian
Formulations for continuous systems and fields:Transition from discrete to continuous system,
Lagrangian formulation for continuous systems stress- energy tensor and conservation theorems.
Hamiltonian formulation otherstheorems

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Tutorial(15 Hours) one hour per week