Problems involving Calculus of Variations: Gold-diggers Problem, Catenary,
Brachystochrone, Dido's problem, Geodesics, minimal surface, optimal harvest,
Revision: Extremals in Finite Dim Calculus (Functions of one and several variables), Euler Lagrange equation (E-L eqns)
Week 2: Special cases E-L eqns: (1) Functions depending on y',
(2) Functions with no explicit 'x' dependence,
(3) Functions with no explicit 'y' dependence,
(4) degenerate functions.
Invariance of E-L eqns, existence, uniqueness of solutions,
Week 3: Generalization : (1) Functionals containing higher derivatives
Generalization: (2) Functionals containing several dependent variables,
(3) Functionals containing two independent variables.
Numerical solution: (1) Euler's FD Method, (2) Ritz Method, (3) Kantorovich's Method
Isoperimetric Problem: Finite dim case/ Lagrange Multipliers including (a) single constraint, (b) multiple constraints, (c) Abnormal problems. Isoperimetric Problems involving functional including cases of generalization in higher dimension, multiple isoperimetric constraints, several dependent variables
Week 5:Holonomic and non-Holonomic Constraints, Problems with Variable endpoints: Natural BCs, Solution of Elastica
Week 6:Problems with Variable endpoints: case of several dependent variable, Transversality conditions, Broken extremals (Weierstrass Erdmann Condition), Newton's Aerodynamic Problem. Hamiltonian formulation of E-L Eqns.
Week 7: Hamiltonian formulation: Case of several dependent variables, Symplectic Maps, Hamilton-Jacobi Equations (HJ Eqns), Method of seperation of variables for HJ Eqns.
Variational Symmetries, Noether's Theorem, Finding Variational Symmetries. Second Variation: Finite dim case, Legendre Condition
Week 9:Conjugate points, Jacobi necessary condition, Jacobi Accessory Eqns (JA Eqns), Sufficient Conditions, finding Conjugate points, saddle points. Optimal Control Theory (OC): solving OC systems via Variational Techniques
Week 10:OC Theory: Constrained Optimization, Pontrygin Minimum Principle (PMP), Hamilton-Jacobi-Bellmann Eqns (HJB), Penalty function method, Slack Variable Method.
Week 11: Nanomechanics: Oscillatory motion of Carbon Nanotube (CNT), Basics (special functions): Pochammer symbol, Hypergeometric Function (HF). Basics (Physical Chemistry): van der Waal Interaction Energy, Lennard Jones Potential. Oscillatory Motion of DWCNT via Hamilton's Principle
Week 12:Additional problem solving sessions.