Your Custom Text Here

This is a one semester course given to the senior undergraduate students at the University of Mumbai- Department of Atomic Energy (UM-DAE) Centre for Excellence in Basic Science. The course content is as follows:

Course Outline :

  1. Variational Principle and derivation of Euler Lagrange Equation, D'Alembert Principle of Virtual Work.

  2. Symmetry and Conservation Laws, Noether's Theorem.

  3. Motion under Central Forces - Derivation of Kepler's laws and orbits.

  4. Motion of Rigid Bodies, Moment of Inertia Tensor, Rotation about a fixed axis, Symmetric Top, Euler Angles.

  5. Small Oscillations, Normal Modes

  6. Hamilton's Equations

  7. Hamilton-Jacobi Equations

Text / References

  1. L. D. Landau and E. M. Lifshitz : A course in Theoretical Physics - Vol. 1, Mechanics, Elsevier (Indian Reprint - 2010)

  2. H. Goldstein, C. Poole and J. Safko, Classical Mechanics (Third Edition), Addison Wesley

 

Course Content

Lecture 1: Variational Principle and Euler Lagrange Equations: Euler-Lagrange Equation is derived from variational principle, introduces generalized coordinates and velocities, degrees of freedom, Application to Brachistochrone problem.

Lecture 2: de' Alembert's Principle and derivation of Euler Lagrange Equations: Principle of virtual work is discussed, Extension to de' Almbert's principle, Generalized force, Inclusion of Lorentz force in a velocity dependent potential.

Problem Set 1 (Assignment): Problems on Euler Lagrange Equations

Lecture 3: Noether's Theorem: Continuous Symmetry of Lagrangian and Noether's Theorem

Problem Set 2: Problems on Symmetry of Lagrangian

Lecture 4: Central Forces: General properties of central force, Kepler's problem, Runge-Lenz Vector, Obtaining force law from orbit equation, Advancement of perihelion of Mercury.

Problem Set 3: Problems Related to Central Forces

Lecture 5: Small Oscillations and Normal Modes: Matrix formulation of coupled oscillator problems, Damped and Forced oscillations, Rayleigh Dissipation function

Problem Set 4: Problems related to small oscillations and normal modes

Lecture 6: Hamiltonian Formalism: Hamilton's equations of motion are derived and several problems discussed

Problem Set 5: Problems on Hamilton's equations

Notes on Lagrange Multipliers

Rigid Body Dynamics: Expressions for velocity and acceleration in a rotating frame are obtained. Non-inertial forces are discussed with emphasis on Coriolis forces, Moment of Inertia tensor introduced and Euler equations are derived. Euler angles are defined and angular velocity vector is discussed. Dynamics of a symmetric top is discussed.

Problem Set 6: Problems on rigid body motion

Notes on Canonical Transformation

Problem Set 7: Problems on Canonical Transformation

Hamilton Jacobi Equations: Hamilton's principal function, Hamilton Jacobi Equation, Action and Angle variables

Problem Set 8: Problems on Hamilton-Jacobi equations