Classical Mechanics

Syllabus Details

• Unit I:(Curvilinear Coordinates and Particle Mechanics) Curvilinear co-ordinates: Cylindrical and spherical polar co-ordinates. Mechanics of a particle, mechanics of a system of particles, types of constraints, d’Alembert’s principle.
• Unit II:(Lagrangian Dynamics and Variational Principles) Lagrange’s equations, Lagrangian formulation in generalized co-ordinates, variational principles, Hamilton’s principle of least action, derivation of Lagrange’s equations from Hamilton’s principle.
• Unit III:(Hamiltonian Mechanics and Canonical Transformations) Legendre transformation, Hamiltonian, canonical equations, cyclic coordinates, Routh’s procedure, generating functions, Poisson brackets, Liouville’s theorem, infinitesimal canonical transformations, conservation theorems and angular momentum relations using Poisson brackets.
• Unit IV:(Hamilton-Jacobi Theory) Hamilton-Jacobi equations, Hamilton’s Principal Function, example of Harmonic oscillator, Hamilton-Jacobi equation for Hamilton’s Characteristic Function, action-angle variables in systems of one degree of freedom.
• Unit V:(Central Force Motion and Moving Frames) Central force problem: Equations of motion and first integrals, equivalent one dimensional problem, classification of orbits; Kepler’s problem: inverse square law of force. Moving frames of reference: Non-inertial frames of reference, rate of change of a vector in a rotating frame, applications to particle kinetics, motion relative to earth, effects of Coriolis.
• Unit VI:(Kinematics of Rigid Bodies) Two dimensional problems in rigid body dynamics, examples. Kinematics of rigid body motion: Euler angles, Euler’s theorem on the motion of a rigid body, infinitesimal rotations.
• Unit VII:(Rigid Body Dynamics and Equations of Motion) The rigid body equations of motion: Angular momentum and kinetic energy of motion, inertia tensor and moment of inertia, inertial ellipsoid, the eigenvalues of the inertia tensor and the principal axis transformation, Euler’s dynamical equations of motion under no external forces, torque-free motion of a rigid body, heavy symmetrical top with one point fixed.

Course Learning Objective

1. Explain the basic concepts of classical Mechanics, apply Newton’s laws and write the kinetic energy and potential energy of a system in terms of generalized coordinates.
2. Distinguish the Lagrangian and Hamiltonian approaches to solve the equations of motion, rigid body dynamics.
3. Use Poisson brackets to determine whether a given transformation is canonical or not.
4. Distinguish between the inertial and non-inertial frames of reference and write the rate of change of vector quantities in rotating frames.

Recommended Books

1. Goldstein, H., Poole, C. P. and Safko, J., Classical Mechanics, third edition, Pearson, 2011.
2. Chorlton, F., Textbook of Dynamics, second edition, Ellis Horwood Series: Mathematics and its Applications, Halsted Press (John Wiley & Sons, Inc.), New York, 1983.
3. Marion, J. B. and Thornton, S. T., Classical Dynamics of Particles and Systems, third edition, Harcourt Brace Jovanovich, 1988.
4. Scheck, Florian, Mechanics. From Newton’s Laws to deterministic Chaos, fifth edition, Graduate Texts in Physics, Springer, Heidelberg, 2010.
5. Marsden, Jerrold E. and Ratiu, Tudor S., Introduction to Mechanics and Symmetry, A Basic Exposition of Classical Mechanical Systems, Texts in Applied Mathematics, 17, Springer-Verlag, New York, 1994.
6. Jose, Jorge V. and Saletan, E. G., Classical Dynamics, A Contemporary Approach, Cambridge University Press, 1998.