Background: During the pandemic, I recorded my online lectures and shared the videos with my students. I had students living in various parts of India and some of them had very poor connectivity. The size of the videos was kept small through post-processing. These 40 lectures on Classical Mechanics were given during August-November 2021. While the lectures are not perfect, I think they should be useful to students interested in learning Classical Mechanics at the M.Sc. level. Students already had one course during their B.Sc. albeit at a lower level.
The Lectures
Pre-requisites: A first course on Classical Mechanics — some familiarity with Newton's equations, solving simple differential equations etc. It is imperative that you do all the problem sets as they are an integral part of the course.
Tips on watching the lectures: Watch the videos at 1.25X slowing it down when necessary. Many modules work as standalone sets of lectures with some weak dependence on earlier lectures.
Slides for the Introductory Lecture Contents: Syllabus and references, A short discussion on the scientific method.
| Module 1: Vectors, Tensors and all that |
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| Lecture Number | Content of the Lecture | Additional Info | Problem Sets/Other Links |
|---|---|---|---|
| Lecture 1 | An introduction to orthogonal matrices | Slides for Lecture 1 | |
| Lecture 2 | Scalar, Vectors and Tensors; Covariance and Invariance of equations | Slides for Lecture 2 | |
| Lecture 3 | The Galilean group and the formal definition of a group. | Problem Set 1 | Slides for Lecture 3 |
Fun reading:
- This article describes how Maxwell's equations were written in a form where covariance under rotations became manifest. J. C. Rautio, "The Long Road to Maxwell's Equations," in IEEE Spectrum, vol. 51, no. 12, pp. 36-56, December 2014, doi: 10.1109/MSPEC.2014.6964925.
- On the Galilean invariance of the Schrodinger equation
| Module 2: Dynamical Systems |
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| Lecture 4 | An introduction to Dynamical Systems | Slides for Lecture 4 | |
| Lecture 5 | Motion in one dimension | Slides for Lecture 5 | |
| Lecture 6 | Motion in various dimensions | Problem Set 2 | Slides for Lecture 6 |
| Module 3: The action principle in classical mechanics |
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| Lecture 7 | Functionals and their extrema | Slides for Lecture 7 | |
| Lecture 8 | Functional Calculus | Slides for Lecture 8 | |
| Lecture 9 | Taylor Series for Functionals | Problem Set 3 | Slides for Lecture 9 |
| Lecture 10 | Symmetries and Constants of Motion | Slides for Lecture 10 | |
| Lecture 11 | Modelling Systems (Statics) | Slides for Lecture 11 | |
| Lecture 12 | Modelling Systems (Dynamics) | Problem Set 4 | Slides for Lecture 12 |
Additional Reading:
- V. Balakrishnan, All about the Dirac Delta Function, Resonance, Aug 2003
- K Lodha, A Roy and S Kar Physics of a Particle on a Rotating Hoop: Experiment and Theory Resonance 25 Issue 9 (2020) pp 1261-1281
| Module 4: Central Forces |
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| Lecture 13 | Motion under a central force | Slides for Lecture 13 | |
| Lecture 14 | The Kepler Problem (Part 1 of 2) | Slides for Lecture 14 | |
| Lecture 15 | The Kepler Problem (Part 2 of 2) | Slides for Lecture 15 |
Additional Reading: V. Balakrishnan, S. Govindarajan and S. Lakshmibala, The central force problem in n dimensions Resonance 25 Issue 4 (2020) pp 513-538
| Module 5: Hamiltonian Methods |
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| Lecture 16 | The Hamiltonian approach to Classical Mechanics, | Slides for Lecture 16 | |
| Lecture 17 | Hamiltonian Mechanics: Poisson brackets, Constants of motion | Problem Set 5 | Slides for Lecture 17 |
| Lecture 18 | Canonical Transformations (Part 1 of 2) | Guest Lecture by Dr. Sunethra Ramanan | Slides for Lecture 18 |
| Lecture 19 | Canonical Transformations Part (2 of 2) | Problem Set 6 | Slides for Lecture 19 |
| Lecture 20 | Hamilton-Jacobi equations | Slides for Lecture 20 | |
| Lecture 21 | Integrable and non-integrable systems | Guest Lecture by Prof. Arul Lakshminarayan | Slides for Lecture 21 |
| Lecture 22 | Action-angle variables | Problem Set 7 | Slides for Lecture 22 |
- You are now ready to read this article in Resonance (Vol 16 (2011) pp 129-151) titled Symmetries and Conservation Laws in Classical and Quantum Mechanics - Classical Mechanics authored by K S Mallesh, S Chaturvedi, V Balakrishnan, R Simon and N Mukunda. This is part one of a two part series of articles. The second part deals with quantum mechanics.
- Another all-time favourite of mine is the article by Dothan titled Finite-Dimensional Spectrum-Generating Algebras that appeared in Phys. Rev. D 2 (1970) 2944
| Module 6: Non-inertial frames |
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| Lecture 23 | Non-inertial frames (Part 1 of 2) | Slides for Lecture 23 | |
| Lecture 24 | Non-inertial frames (Part 2 of 2) | Slides for Lecture 24 | |
| Lecture 25 | The Foucault Pendulum | Problem Set 8 | Slides for Lecture 25 |
| Module 7: Dynamics of Rigid Bodies |
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| Lecture 26 | Rigid Body — Statics | Slides for Lecture 26 | |
| Lecture 27 | Rigid Body — Dynamics | Slides for Lecture 27 | |
| Lecture 28 | Dynamics of a symmetric top (Part 1 of 2) | Problem Set 9 | Slides for Lecture 28 |
| Lecture 29 | Dynamics of a symmetric top (Part 2 of 2) | Slides for Lecture 29 |
| Module 8: Small Oscillations |
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| Lecture 30 | Small Oscillations (Part 1 of 3) | Slides for Lecture 30 | |
| Lecture 31 | Small Oscillations (Part 2 of 3) | Problem Set 10 | Slides for Lecture 31 |
| Lecture 32 | Small Oscillations (Part 3 of 3) | Slides for Lecture 32 |
| Module 9: Lie Algebras and Lie Groups |
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| Lecture 33 | Lie Algebras and Lie Groups (Part 1 of 2) | Slides for Lecture 33 | |
| Lecture 34 | Lie Algebras and Lie Groups (Part 2 of 2) | Problem Set 11 | Slides for Lecture 34 |
| Module 10: An introduction to Special Relativity |
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Lectures 35/36/37 is something I came up with around 2009 and got to refine it over the years. Hope you like my uncoventional introduction to special relativity.
| Lecture 35 | A prelude to Special Relativity (Part 1 of 2) | Slides for Lecture 35 | |
| Lecture 36 | A prelude to Special Relativity (Part 2 of 2) | Slides for Lecture 36 | |
| Lecture 37 | The Lorentz Group (Part 1 of 2) | Fixing the three constants | Slides for Lecture 37 |
| Lecture 38 | The Lorentz Group (Part 2 of 2) | Slides for Lecture 38 | |
| Lecture 39 | Tensors of the Lorentz Group | Slides for Lecture 39 | |
| Lecture 40 | Mechanics in Special Relativity | Problem Set 12 | Slides for Lecture 40 |
Must Watch: This is an educational video the describes in detail the 1960 experiment of Frisch and Smith that tested the SR prediction of time dilation (as discussed in Lecture 40). The ingenuity of the experiment is something to behold.
Time Dilation: An experiment with muons The results of the experiment is described here: Measurement of the Relativistic Time Dilation Using μ-Mesons American Journal of Physics 31, 342 (1963)
Fun stuff: English Translation of Einstein's 1905 paper on SR