Syllabus (PHYSICS)

Course Type: MAJ-14

Semester: 8

Course Code: BPHSMAJ14T

Course Title: Classical and Relativistic Mechanics

(L-P-Tu): 3-0-1

Credit: 4

Practical/Theory: Combined

Course Objective: This course aims to provide a strong foundation in classical mechanics and its extension to relativistic dynamics. It covers constraints, generalized coordinates, Lagrangian and Hamiltonian formulations, variational principles, canonical transformations, and Hamilton–Jacobi theory with applications to oscillatory systems, rigid body motion, and symmetries. The dynamics of rigid bodies, including Euler’s equations, inertia tensor, and rotational motion, are explored alongside relativistic concepts such as Lorentz transformations, time dilation, length contraction, energy–momentum relations, and Minkowski space–time. By integrating analytical methods of mechanics with relativity, the course equips students with essential tools for modern theoretical physics.

Learning Outcome: This course aims to provide a strong foundation in classical mechanics and its extension to relativistic dynamics. It covers constraints, generalized coordinates, Lagrangian and Hamiltonian formulations, variational principles, canonical transformations, and Hamilton–Jacobi theory with applications to oscillatory systems, rigid body motion, and symmetries. The dynamics of rigid bodies, including Euler’s equations, inertia tensor, and rotational motion, are explored alongside relativistic concepts such as Lorentz transformations, time dilation, length contraction, energy–momentum relations, and Minkowski space–time. By integrating analytical methods of mechanics with relativity, the course equips students with essential tools for modern theoretical physics.

Classical and Relativistic Mechanics (4 Credits)

Theory (4 Credits)

Constraints and Constrained Motion: Constraints and their classification, virtual displacement and virtual work, D’Alembert’s principle and its application, Principle of virtual work, Degrees of freedom, Generalized coordinates and velocities. holonomic and nonholonomic systems.

[7 Lectures]

Lagrangian Mechanics: Elements of the calculus of variations. Stationary value of a definite integral. The brachistochrone problem. Hamilton’s variational principle. Derivation of Lagrange’s equations of motion from Hamilton’s principle. Lagrange’s equations for holonomic systems. Application of Lagnangian formalism to simple systems. Lagrange’s undetermined multipliers, Lagrange’s equation for non-holonomic systems, Cyclic coordinates, Virial theorem, Small oscillations and Normal modes. (13 Lectures)

Hamiltonian Mechanics: Hamilton’s function and Hamilton’s equations of motion. Hamilton’s equations of motion for holonomic systems. Application of Hamiltonian formalism to simple problems. (8 Lectures)

Canonical transformations: Conditions for transformation to be canonical. Lagrange and Poisson brackets as canonical invariants. Equations of motion in Poisson bracket notation. Infinitesimal contact transformation. Constants of the motion. Symmetry properties. Poisson bracket relations. Liouville’s theorem. (7 Lectures)

Hamilton-Jacobi theory: The Hamilton Jacobi equation for Hamilton's principle function. The harmonic oscillator problem. (2 Lectures)

Rigid Body Dynamics: Independent coordinates. Orthogonal transformations and rotations (finite and infinitesimal). Euler’s theorem, Euler angles. Inertia tensor and principal axis system. Euler’s equations. Heavy symmetrical top with precession and nutation. (7 Lectures)

Special Theory of Relativity:

Michelson-Morley Experiment and its outcome; Postulates of Special Theory of Relativity; Lorentz Transformations; Relativistic addition of velocities; Length contraction; Time dilation; Relativistic transformation of velocity, frequency and wave number; Concept of zero rest mass of photon - Variation of mass with velocity; Mass-energy equivalence.

Minkowski space: invariant interval, light cone and world lines; Four-vectors: space-like, time-like and light-like; Four-velocity and acceleration; Four-momentum and energy-momentum relation; Lorentz transformations of four-vectors; Metric and alternating tensors; Relativistic dynamics and kinematics; Lagrangian and Hamiltonian of a relativistic particle. (16 Lectures)

Reading References

Theory

1. Classical Mechanics, H. Goldstein, C.P. Poole, and J.L. Safko, Pearson, 3^rd Edn.
2. Introduction to Classical Mechanics, D. Morin, Cambridge University Press
3. Classical Mechanics, P. S. Joag, N. C. Rana, Tata McGraw Hall.
4. Theoretical Mechanics, M. R. Spiegel, Schaum’s Outline Series, McGraw-Hill.
5. Classical Mechanics, J. C. Upadhyaya, Himalaya Publishing House, 2^nd Edn.
6. Fundamentals of Classical Mechanics, A.B. Gupta, Books & Allied P vt Ltd
7. Introduction to Special Relativity, R. Resnick, John Wiley and Sons.
8. Special Relativity, A. P. French, Viva Books.
9. The Special Theory of Relativity, S Banerji and A Banerjee, PHI, 2^nd Edn.