Unit I - Lagrangian formulation (14 hrs)
1.1 Review of Newtonian Mechanics: Mechanics of a Particle; Mechanics of a System of Particles; Constraints.
1.2 D' Alembert's principle and Lagrange’s equations; velocity-Dependent potentials and the Dissipation Function; Lagrangian for a charged particle in electromagnetic field.
1.3 Application of Lagrange’s equation to: motion of a single particle in Cartesian coordinate system and plane polar coordinate system; bead sliding on a rotating wire.
1.4 Hamilton’s Principle; Technique of Calculus of variations; The Brachistochrone problem.
1.5 Derivation of Lagrange’s equations from Hamilton’s Principle.
1.6 Canonical momentum; cyclic coordinates; Conservation laws and Symmetry properties—homogeneity of space and conservation of linear momentum; isotropy of space and conservation of angular momentum; homogeneity of time and conservation of energy; Noether’s theorem (statement only; no proof is expected).
Hamiltonian formulation: (4 hrs)
1.7 Legendre Transformations; Hamilton’s canonical equations of motion; Hamiltonian for a charged particle in electromagnetic field.
1.8 Cyclic coordinates and conservation theorems; Hamilton’s equations of motion from modified Hamilton’s principle.
Unit II - Small oscillations (8 hrs)
2.1 Stable equilibrium, unstable equilibrium, and neutral equilibrium; motion of a system near stable equilibrium—Lagrangian of the system and equations of motion.
2.2 Small oscillations—frequencies of free vibrations; normal coordinates and normal modes.
2.3 System of two coupled pendula—resonant frequencies, normal modes, and normal coordinates; free vibrations of CO2 molecule—resonant frequencies, normal modes, and normal coordinates.
Canonical transformations and Poisson brackets (10 hrs)
2.4 Equations of canonical transformations; Four basic types of generating functions and the corresponding basic canonical transformations.
2.5 Solution of harmonic oscillator using canonical transformations.
2.6 Poisson Brackets; Fundamental Poisson Brackets; Properties of Poisson Brackets.
2.7 Equations of motion in Poisson Bracket form; Poisson Bracket and integrals of motion; Poisson’s theorem; Canonical invariance of the Poisson bracket.
Unit III - Central force problem (9 hrs)
3.1 Reduction of two-body problem to one-body problem; Equation of motion for conservative central forces—angular momentum and energy as first integrals; law of equal areas.
3.2 Equivalent one-dimensional problem—centrifugal potential; classification of orbits.
3.3 Differential Equations for the orbit; equation of the orbit using the energy method; The Kepler Problem of the inverse square law force; Scattering in a central force field—Scattering in a Coulomb field and Rutherford scattering cross section.
Rigid body dynamics (9 hrs)
3.4 Independent coordinates of a rigid body; Orthogonal transformations; Euler Angles.
3.5 Infinitesimal rotations: polar and axial vectors; rate of change of vectors in space and body frames; Coriolis effect.
3.6 Angular momentum and kinetic energy of motion about a point; Inertia tensor and the Moment of Inertia; Eigenvalues of the inertia tensor and the Principal axis transformation.
3.7 Euler equations of motion; force-free motion of a symmetrical top.
Unit IV - Hamilton-Jacobi theory and action-angle variables (12 hrs)
4.1 Hamilton-Jacobi Equation for Hamilton’s Principal Function; physical significance of the principal function.
4.2 Harmonic oscillator problem using the Hamilton-Jacobi method; Hamilton-Jacobi Equation for Hamilton’s characteristic function.
4.3 Separation of variables in the Hamilton-Jacobi Equation; Separability of a cyclic coordinate in Hamilton-Jacobi equation; Hamilton-Jacobi equation for a particle moving in a central force field (plane polar coordinates).
4.4 Action-Angle variables; harmonic oscillator problem in action-angle variables.
Classical mechanics of relativity (6 hrs)
4.5 Lorentz transformation in matrix form; velocity addition; Thomas precession.
4.6 Lagrangian formulation of relativistic mechanics; Application of relativistic Lagrangian to (i) motion under a constant force, (ii) harmonic oscillator, and (iii) charged particle under constant magnetic field.