Mechanics of a system of particles in vector form. Conservation of linear momentum, energy and angular momentum – Degrees of freedom, generalized coordinates and velocities. Lagrangian, action, and the principle of extremal action. Euler-Lagrange equations. Constraints. Applications of the Lagrangian formalism – Legendre transform, generalized momenta. Hamiltonian and Hamilton’s equations of motion. Phase space, phase trajectories. Applications to systems with a small number of degrees of freedom – Scattering in a central potential, Rutherford formula, scattering cross section – Charged particle in an electromagnetic field – Non-inertial frames of reference and pseudo-forces: centrifugal, Coriolis and Euler forces – Rotations in three dimensions, Euler angles. Angular momentum. Elements of rigid-body dynamics. The symmetric top – Small oscillations. Normal mode analysis. Normal modes of simple molecules and harmonic chains – Elementary ideas on general dynamical systems: conservative versus dissipative systems. Hamiltonian systems and Liouville’s theorem – Constants of the motion and symmetries in classical mechanics. Noether’s Theorem and the connection between symmetry, invariance and conservation principles – Canonical transformations, Poisson brackets. Action-angle variables. Integrable Hamiltonians. Non-integrable systems and elements of chaotic motion – Special relativity: Inertial frames. Principle and postulate of Relativity. Lorentz transformations. Length contraction, time dilation and the Doppler effect. Velocity addition formula. Rapidity – Four-vectors and tensors. Energy-momentum of a free particle. Relativistic invariance of physical laws. The wave equation. Transformation properties of electric and magnetic fields. The homogeneous and inhomogeneous Lorentz groups

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This course is delivered by NPTEL, is part of IIT Madras online courses.

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