# Classical Mechanics: From Newtonian to Lagrangian Formulation

By Prof. Debamalya Banerjee   |   IIT Kharagpur
Learners enrolled: 1784
This course deals with fundamentals of classical mechanics. Mechanics is one of the core subjects for physics and engineering disciplines. This course starts from basics of Newtonian mechanics. Then we introduce rigid dynamics and finally Lagrangian formulation of dynamics followed by small oscillations. We also offer tutorials to develop problem solving skills. We aim to give a basic understanding of various fields of classical mechanics to our students.
INTENDED AUDIENCE : UG students from various degree and engineering colleges
Summary
 Course Status : Ongoing Course Type : Core Duration : 12 weeks Start Date : 26 Jul 2021 End Date : 15 Oct 2021 Exam Date : 24 Oct 2021 IST Enrollment Ends : 09 Aug 2021 Category : Physics Credit Points : 3 Level : Undergraduate

### Course layout

Week 1: Review of basic Newtonian mechanics, kinematics problems, Motion under resistance: terminal velocity, System with variable mass: Rocket, raindrop, etc.
Week 2: Central forces: plane polar co-ordinate system,2-body central force problem, general equation of the orbit, Kepler’s laws of planetary motion
Week 3: Effective potential of central force, Escape velocity, eccentricity of the orbit under various initial conditions Satellite, Ballistic missile and orbit transfer
Week 4: Moving co-ordinate system: pseudo forces, Coriolis and centrifugal force, Foucault’s pendulum, Introduction to system of particles
Week 5: Dynamics in center of mass frame, introduction to rigid body,Degrees of freedom (DOF) and constrains, Rotational dynamics of rigid body, moments of inertia tensor
Week 6: Principle moments of inertia, ellipsoid of inertia, parallel and perpendicular axis theorem, Euler’s equation of rigid body rotation under external torque, Torque free motion of spherical, symmetric and asymmetric top
Week 7: Introduction to Euler angles, pitch, precession and nutation, The heavy symmetric top, the energy equation Special case of ‘fast’ and ‘sleeping’ top
Week 8: Lagrangian dynamics: Forces of constrain, virtual displacement, Principle of virtual wok and D’Alembert’s principle, Generalized co-ordinates, Lagrange’s equation of 1st kind, Lagrange’s undetermined multiplier
Week 9: Generalized velocity and force, Lagrange’s equation of 2nd kind, Lagrangian, Classification of constrains, Lagrange’s equations for non-holonomic systems, Dissipation and gauge function, most general form of Lagrange’s equation, review of rigid body including heavy symmetric top
Week 10: Variation principle, calculus of variation, Principle of least action and Lagrange’s equation from it Application of variation principle (catenary, geodesic,etc.)
Week 11: Small oscillation: Introduction to coupled systems, normal modes, Types of equilibrium, Kinetic and potential energy of a coupled system, expression of energy in matrix form, Equation of motion of a coupled system, normal frequencies , eigenvalues and eigenvectors of K.E. and P.E. matrices,
Week 12: Diagonalization of K.E. and P.E. matrices, Normal modes of oscillation, Examples (linear molecules, spring mass systems etc.)

### Books and references

1. Spiegel, M. R. (1967). Theory and Problems of Theoretical Mechanics, Schaum’s Outline Series.
2. Raychaudhuri, A. K. (1983). Classical mechanics: a course of lectures. Oxford University Press.
3. Goldstein, H. (1965). Classical mechanics. Pearson Education India.
4. Raychaudhuri, D. P. & Maiti, S. N. (2006) Classical Mechanics & General Properties of Matter, New    age International (P) Limited
5. Landau, L. D., & Lifshitz, E. M. (1976). Mechanics, vol. 1. Course of theoretical physics, Pergamon Press

### Prof. Debamalya Banerjee

IIT Kharagpur
Prof. Debalmalya Banerjee is an Assistant Professor at department of physics, IIT Kharagpur. I am taking classical mechanics for 2nd year UG students of the department for past 3 years.

### 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 October 2021 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 Kharagpur. 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