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# Variational Calculus and its applications in Control Theory and Nanomechanics

## Overview

This course is designed as an introduction to the theory and applications of Variational Calculus to problems in geometry, differential equations and physics, particularly mechanics. This course assumes very limited knowledge of vector calculus, ordinary differential equations and basic mechanics. Many new applications in applied mathematics, physics, chemistry, biology and engineering are included. This course will serve as a reference for advanced study and research in this subject as well as for its applications in the fields of industrial control systems and instrumentation engineering, nanoscience and software development.
INTENDED AUDIENCE :
NonePREREQUISITES : 1) Multivariable Calculus 2) Ordinary Differential Equations (optional)INDUSTRIES SUPPORT :Industries in areas of (1) Control System and Instrumentation Engineering, (2) Nanoscience, (3) Software Development

## Syllabus

### COURSE LAYOUT

Week 1:Introduction
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,
Generalization : (1) Functionals containing higher derivativesWeek 3: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 MethodWeek 4: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 ElasticaWeek 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.Week 8: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 TechniquesWeek 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 PrincipleWeek 12:Additional problem solving sessions.

### Taught by

Prof. Sarthok Sircar

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