Classical Mechanics

Course Summary

The course provides a solid foundation in Classical Mechanics. We will begin by formulating the primary concepts: the laws of motion and the dynamical equations as differential equations which need to be integrated to find the solutions of mechanical problems. We will stress the importance of conservation laws not only in the solution of otherwise intractable dynamical problems but also their wider scope beyond Newtonian dynamics. As applications we will deal with central force motion and the harmonic oscillator. Besides their historical importance, these are rich and instructive examples illustrating the basic concepts.

The second part of the course deals with Analytical Mechanics: We will make the transition from newtonian to lagrangian dynamics by introducing the notion of generalised co-ordinates. The variational principle is then introduced as a unified way of dealing with diverse physical problems. Lagrange’s equations are re-derived and the transition to the hamiltonian form of dynamics is made thereby introducing the concept of phase space - a notion of special importance in statistical mechanics. Subsequently, the Poisson bracket formulation would be discussed. Throughout we will stress the importance of, and the connections between, symmetry principles and conservation laws.

This course would be a stepping stone for later courses in quantum and statistical mechanics. It should also enable you to later study more advanced topics in classical physics like fluid mechanics, non-linear dynamical systems and chaos.

Course Outline

Space and Time, Force and Inertia, Inertial frames, Newton’s Laws, equations of motion, Galilean transformations as symmetries - principle of relativity. Work- Energy Theorem, Kinetic and Potential Energy. Rotational motion: Torque, angular momentum and moment of inertia.

Harmonic Oscillator: simple, damped and driven oscillators.
Many particle systems: Energy, Momentum and Angular Momentum - their conservation laws and use in solving dynamical problems.

Constraints, generalised coordinates and configuration space, the transition from Newtonian to Lagrangian Mechanics, ignorable coordinates.
The Principle of Least Action: motivation, Fermat’s principle of least time, the Brachistochrone problem, general formulation and Euler-Lagrange equations of motion. Symmetries and conservation laws - Noether’s theorem.

Law of Gravitation, Kepler’s Laws and Planetary Orbits: motion in a Central force. LRL vector and its use in deriving the orbit equation.

The Legendre transform and Hamiltonian function, Hamilton’s equations of motion, phase space, Liouville’s theorem, Poisson brackets and connections to quantum mechanics.

Non-Inertial frames of reference, effects of acceleration and gravity, the principle of equivalence of inertial and gravitational mass as the foundation of the general theory of relativity.

Books:
Recommended preparatory/supplementary reading: Feynman lectures on Physics Vol. 1, upto chapter 21

Marion and Thornton, Classical Dynamics of particles and systems [Cengage]
R. Douglas Gregory, Classical Mechanics [CUP]. These two texts cover most of the lectured material at essentially the same level.

Landau and Lifshitz, Mechanics [Elsevier]