# Variational Calculus and its applications in Control Theory and Nanomechanics

By Prof. Sarthok Sircar   |   IIIT Delhi
Learners enrolled: 299
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
None
PREREQUISITES : 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
Summary
 Course Status : Completed Course Type : Elective Duration : 12 weeks Category : Mathematics Credit Points : 3 Level : Undergraduate/Postgraduate Start Date : 24 Jan 2022 End Date : 15 Apr 2022 Enrollment Ends : 07 Feb 2022 Exam Date : 24 Apr 2022 IST

Note: This exam date is subjected to change based on seat availability. You can check final exam date on your hall ticket.

### 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 derivatives
Week 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 Method
Week 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 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.
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 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

### Books and references

1) Calculus of variations: Bruce. Van Brunt
2) Optimal Control Theory: D. S. Naidu
3) Problems and Exercises in CoV: Krasnov, Makarenko, Kiselev
4) Calculus of Variations: Gelfand, Formin, Silverman
5) Calculus of Variations: Craigs,
6) Calculus of Variations: Aurther, Routledge, Paul
7) A primer on Calculus of Variations and Optimal Control Theory: Gibbons
8) Handbook of Mathematical Functions: Abramowitz & Stegun

### Prof. Sarthok Sircar

IIIT Delhi
I am currently an Assistant Professor in the Department of Mathematics at Indraprastha Institute for Information Technology, Delhi. My prior academic appointments include (a) Lectureship in the Division of Mathematical Sciences, University of Adelaide, Australia, (b) Research Associate in Division of Applied Mathematics, University of Colorado, Boulder, (c) Research Fellow in Biomathematics in the University of Utah, (d) Visiting Scholar in the Center for Nanophase Material Science at Oak Ridge National Laboratory, and (e) Research Scientist in Corning Inc. at Ithaca, NY. My main mathematical interests are in the development and analysis of nonlinear hyperbolic and elliptic partial differential equations, with applications which lie at the interface of applied mathematics and biology. I am particularly interested in solving problems involving soft matter and fluid flow using asymptotic and perturbation methods, numerical approximation and statistical mechanics.

### Course certificate

The course is free to enroll and learn from. But if you want a certificate, you have to register and write the proctored exam conducted by us in person at any of the designated exam centres.
The exam is optional for a fee of Rs 1000/- (Rupees one thousand only).
Date and Time of Exams: 24 April 2022 Morning session 9am to 12 noon; Afternoon Session 2pm to 5pm.
Registration url: Announcements will be made when the registration form is open for registrations.
The online registration form has to be filled and the certification exam fee needs to be paid. More details will be made available when the exam registration form is published. If there are any changes, it will be mentioned then.
Please check the form for more details on the cities where the exams will be held, the conditions you agree to when you fill the form etc.

CRITERIA TO GET A CERTIFICATE

Average assignment score = 25% of average of best 8 assignments out of the total 12 assignments given in the course.
Exam score = 75% of the proctored certification exam score out of 100

Final score = Average assignment score + Exam score

YOU WILL BE ELIGIBLE FOR A CERTIFICATE ONLY IF AVERAGE ASSIGNMENT SCORE >=10/25 AND EXAM SCORE >= 30/75. If one of the 2 criteria is not met, you will not get the certificate even if the Final score >= 40/100.

Certificate will have your name, photograph and the score in the final exam with the breakup.It will have the logos of NPTEL and IIT Madras .It will be e-verifiable at nptel.ac.in/noc.

Only the e-certificate will be made available. Hard copies will not be dispatched.

Once again, thanks for your interest in our online courses and certification. Happy learning.

- NPTEL team