Quick Facts

Medium Of Instructions	Mode Of Learning	Mode Of Delivery
English	Self Study	Video and Text Based

Course Overview

The online Variational Calculus and its applications in Control Theory and Nanomechanics training course has been designed as an introductory session to the main theory and applications that are related to variational calculus to problems with respect to- physics, geometry, and differential equations along with mechanics. This course on Variational Calculus and its applications in Control Theory and Nanomechanics by Swayam provides the students with adequate knowledge on the topics of- ordinary differential equations, vector calculus, and basic mechanics. In the course tenure, the applicants will be learning about the new applications that are being applied in- physics, biology, mathematics, chemistry, and engineering.

The Variational Calculus and its applications in Control Theory and Nanomechanics certification benefit the students as this will serve as a reference for advanced study and research in the same domain. After completing the course the students will have their skills increased in the field of functionals that contains several dependent variables.

The Highlights

Course offered by Swayam
Course partnered by IIIT Delhi
12 weeks program
The certificate recognized by the government
Course type- elective
AICTIE approved FD course

Programme Offerings

audio lectures
video lectures
Reading Material
references
Books
Online discussion forum
online learning
assessments
assignments

Courses and Certificate Fees

Fees Informations	Certificate Availability	Certificate Providing Authority
INR 1000	yes	IIIT Delhi

Variational Calculus and its applications in Control Theory and Nanomechanics fee for the registration is kept free.
Rs.1,000/- has to be paid by the students to apply for the examination.

Particulars	Amount
Registration fee	Free
Examination fee	Rs.1,000/-

Eligibility Criteria

Education

It is necessary that the students are having a basic understanding of- ordinary differential equations (optional) or multivariable calculus before they apply for the Variational Calculus and its applications in Control Theory and Nanomechanics program.

Certification Qualification Details

The Variational Calculus and its applications in Control Theory and Nanomechanics certification will be only offered to candidates who will register for examination by paying a fee and thereafter secure a 75% minimum in it. Also, they should also get a 25% average in 8 out of 12 assignments. Thereafter they will be awarded Variational Calculus and its applications in Control Theory and Nanomechanics online courses.

What you will learn

Mathematical skill

In the Variational Calculus and its applications in Control Theory and Nanomechanics training, the following topics will be covered-

The students will be introduced to the problems that involve calculus as a concept.
They will be learning about the functions that pertain to X dependence.
Candidates will have their skills improved on generalization.
In the Variational Calculus and its applications in Control Theory and Nanomechanics certification syllabus, the students will cover isoperimetric problems.
Students will further be learning about the independent variables.
Candidates will also be taught about non-holonomic and holonomic functions.
They will be solving questions that are related to variable endpoints.
Applicants will also be taught about Lagrange multipliers.
The registered students will be covering the chapters of a single constraint.
Candidates in the course will further study the solutions of elastia

Who it is for

The Variational Calculus and its applications in Control Theory and Nanomechanics online course is for-

Students who want to secure a career in the field of software development can apply for this course.

Application Details

For securing the seat in the Variational Calculus and its applications in Control Theory and Nanomechanics program the students have to visit the given link-

Step 1: The mentioned URL has to be visited by the candidates-https://onlinecourses.nptel.ac.in/noc21_ma14/preview

Note- Since, the course is currently being held therefore, the program application is not available currently.

The Syllabus

Introduction

Problems involving Calculus of Variations: Gold-diggers Problem, Catenary,
Brachistochrone, 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)

Special cases E-L eqns

Functions depending on y',
Functions with no explicit 'x' dependence,
Functions with no explicit 'y' dependence,
Degenerate functions.
Invariance of E-L eqns, existence, uniqueness of solutions,
Generalization: Functionals containing higher derivatives

Special cases E-L eqns

Functionals containing several dependent variables,
Functionals containing two independent variables.
Numerical solution: Euler's FD Method, Ritz Method, 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 a higher dimension, multiple isoperimetric constraints, several dependent variables

Holonomic and non-Holonomic Constraints
Problems with Variable endpoints: Natural BCs, Solution of Elastica

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

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

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

OC Theory

Constrained Optimization
Pontrygin Minimum Principle (PMP)
Hamilton-Jacobi-Bellmann Eqns (HJB)
Penalty function method, Slack Variable Method

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

Additional problem solving sessions

Instructors

IIIT Delhi Frequently Asked Questions (FAQ's)

1: After completing the course which industries can I apply for?

The students can apply for the listed industries- Control System and Instrumentation Engineering, Software Development, and Nanoscience.

2: What is the application processing fee?

The students need not pay an amount as the Variational Calculus and its applications in Control Theory and Nanomechanics fee for the application.

3: For the course is scholarship provided?

No, in the Variational Calculus and its applications in Control Theory and Nanomechanics online course no scholarship will be issued.

4: For how many weeks will the program continue?

The course on Variational Calculus and its applications in Control Theory and Nanomechanics training will be organized for twelve weeks.

5: By whom is the certificate going to be issued?

The students will get the certificate from the Institute of NPTEL and IIIT Delhi.

6: Is the course integrated?

No, the course on Variational Calculus and its applications in Control Theory and Nanomechanics online course is an independent course.

7: Will placement assistance be provided?

No, for the Variational Calculus and its applications in Control Theory and Nanomechanics training placement assistance will not be provided.

8: Will the exam be organized online?

No, the examination on the Variational Calculus and its applications in Control Theory and Nanomechanics certification course will take place from an exam center.

9: Where can I access the eligibility details?

The students can access the details for eligibility from the official site of Swayam.