AMAT 42463: Advanced Mechanics

Course Code : AMAT 42463

Title               : Advanced Mechanics

Pre-requisites : AMAT 21272

Learning Outcomes:

After the completion of this course unit, the student will be able to:

1. apply Lagrange’s equations to solve motion involving impulse
2. solve dynamic problems using Hamilton’s equations
3. verify canonical variables, determine generating functions and use canonical transformations
4. use Poisson brackets and their properties for finding constants of motion, determining canonical
transformations
5. approximate problems for small oscillations and determine normal frequencies and normal modes
6. solve motion of a heavy symmetric top by Hamiltonian formulation
7. determine stability of fixed points of a nonlinear dynamical system

Course Contents:

Impulsive motion: 

• Equations of motion, body acted on by a given impulse, body acted on by an impulsive couple, impact of inelastic bodies, elastic bodies, applications of principle of virtual work, Lagrange's equations for impulsive motion

Hamiltonian dynamics: 

• Generalized Momentum, Hamilton's equations of motion, Liouville's Theorem, Poisson brackets, Canonical transformation, Hamilton-Jacobi equation

Small oscillations: 

• Lagrange's method, Normal modes, Roots of the Lagrangian determinant, Oscillations under constraint, Stationary property of the normal modes

The Motion of a Top: 

• Euler's Equations, Free Tops, Euler's Angles, The Heavy Symmetric Top

Chaos theory: 

• Fixed points and their linear stability, Elements of bifurcation theory, Limit cycles, Synchronization and phase dynamics, Chaos and strange attractors, Cellular automata

Method of Teaching and Learning: A combination of lectures, tutorial discussions and presentations.

Assessment: Based on tutorials, tests, presentations and end of course examination.

Recommended Reading:

1. Rao, A.V. (2011). Dynamics of Particles and Rigid Bodies: A Systematic approach, Cambridge
University Press.
2. Chorlton, F. (2 nd Ed., 2019). Textbook of Dynamics, D. Van Nostrand.
3. Gignoux C & Silvestre-Brac, B. (2014). Solved Problems in Lagrangian and Hamiltonian Mechanics,
Springer Netherlands.
4. Kelley, J.D. & Leventhal, J.J. (2016). Problems in Classical and Quantum Mechanics: Extracting the
Underlying Concepts. Springer.
5. Strauch, D. (2009). Classical Mechanics, An Introduction, Springer.
6. Goldstein, H., Poole, C. P. & Safko, J. (3 rd Ed., 2011). Classical Mechanics, Pearson.
7. Mondal, C.R. (2005). Classical Mechanics, Prentice Hall of India.
8. Ramsey, A.S. (1975). Dynamics, Parts I & II, Cambridge University Press.