## Course Description

The main goal of our course is to describe and explain the motion of macroscopic objects acted upon by external forces. The motion could be a translation, rotation, and oscillation. We will use Newtonian, Lagrangian, and Hamiltonian mechanics to model the motion of particles and the system of particles. In addition to the usefulness of these techniques in classical mechanics, they are essential to understanding the foundations of statistical and quantum mechanics. Towards the end of this course, we will spend a few lectures on nonlinear dynamical systems, which would allow us to present a dynamical view of the world.

Due to the prevailing restrictions of covid19, this course will be taught and graded **completely in online mode**. I will record and upload the lectures of a given week – 2 or 3 in total – on the Wednesday of every week. Since the recorded lectures are uploaded, we will arrange a **discussion hour every week on Monday at 14h00 – 15h15**. **Tutorial sessions** to practice problems will take place on **Wednesdays between 14h15 – 15h15**.

- IIT Kanpur’s mooKIT web platform will be used to run the course. All course material will be uploaded on mooKIT website weekly. Students should submit (upload) their home assignments through mooKIT only.
- Assignments will be uploaded every Wednesday before noon and the corresponding solutions should be submitted before noon of the Saturday of the following week. Assignments submitted after 12h00 on Saturday will not be considered for grading.
- Four quizzes will be given, and the dates for the same will be intimated roughly a week before.
- Professional ethics will be given significant importance. If found cheating during the assignments, quizzes, and exams you may expect an
**F**grade.

**Course grading**:

Weightages given to different items.

- Assignments – 10 %
- Quizzes – 20 %
- Mid-semester – 30 %
- End semester – 40 %

Please be informed that I will follow relative grading for the award of final grades. In addition, students with scores below 40% can expect an F grade.

## Course Content

*First part*:- Reviews of Newton’s Laws of motion, Galilean transformations, Frames of references and pseudo forces, Symmetries in Newton’s laws, Lagrangian formulation, Configuration space.
- Calculus of variations, Hamilton’s principle of least action, Euler Lagrange’s equations, Conserved quantities, and Noether’s theorem.
- Small oscillations and normal modes, Anharmonic oscillators, resonance in harmonic and anharmonic oscillators, parametric resonance.
- Secular (regular) perturbation theory, Lindstedt Poincare method, Rigid body dynamics.
- Tutorial Problems: one and two degrees of freedom simple harmonic oscillator, double pendulum, motion in a central force field, the system of particles, charged particle in an electromagnetic field, Lagrangian formulation and relativistic mechanics, etc.

*Second part*:- Fixed points and linear stability analysis, limit cycles, flow on a torus and quasi periodicity, Qualitative discussion on Poincare Bendixon Theorem (no chaos in 2D autonomous flow).
- Legendre transformation, Hamiltonian formulation, Phase plane, Integral variants, symplectic area of conservation, Generalized Liouville’s theorem, Poincare recurrence theorem, Modified Hamilton’s principle.
- Canonical transformations, infinitesimal canonical transformations, Poisson Brackets, Active view vs passive view of canonical transformations.
- Principle of varying action and Hamilton-Jacobi (HJ) theory, Analogy between optics and HJ method in mechanics, Action angle variables.

*Third part*:- Lorenz system, chaotic attractor, Lyapunov exponents.
- A qualitative discussion of nonintegrability, and chaos in Hamiltonian systems.

**Reference Texts**:

- Classical Mechanics, by H. Goldstein.
- Classical Dynamics of Particles and Systems, by S. T. Thornton and J. B. Marion.
- Classical dynamics, by J. V. Jose, and E. J. Saletan.
- Mechanics, by L. D. Landau and L. H. Lifshitz.
- Nonlinear dynamics and chaos, by S. H. Stograz.