Syllabus (PHYSICS)

Course Type: DSC-101

Semester: 1

Course Code: MPHSDSC101T

Course Title: Classical Mechanics

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

Credit: 4

Practical/Theory: Theory

Course Objective: See Below

Learning Outcome: See Below

MPHSDSC101T: Classical Mechanics [4 Credits: 3L + 1T] (45 Lectures + 15 Tutorials)

Course Objective:

• • Gain a deeper understanding of classical mechanics beyond Newton's laws through modern analytical approaches.
  • Learn to formulate and solve complex dynamical problems using the Lagrangian and Hamiltonian methods.
  • Understand how variational principles and the principle of least action provide a unified framework for describing physical systems.
  • Appreciate the connection between symmetry and conservation laws and recognize their fundamental role in physics.
  • Use advanced techniques such as canonical transformations, Poisson brackets and Hamilton–Jacobi theory to simplify and analyze mechanical systems.
  • Develop the ability to study the motion of particles, rigid bodies, oscillatory systems, and systems under central forces with both mathematical rigor and physical insight.
  • Build a strong conceptual foundation for higher-level courses in theoretical and applied physics, including quantum mechanics, electrodynamics, statistical mechanics, plasma physics, nuclear physics, astrophysics, and general relativity.
  • Understand the fundamental principles of the Special Theory of Relativity and apply them to explain the behavior of space, time, energy, momentum, and space–time at relativistic speeds.

Course Outcomes:

• Apart from describing the system classically, the Lagrangian and Hamiltonian approaches will be useful in describing a microscopic system too.
• The concept of Hamilton- Jacobi theory will raise the idea of transition of a system from classical to quantum mechanics.
• The concept of point masses as a whole will be generated by studying the dynamics of a rigid body.
• In addition to classical mechanics, the relativistic mechanics will be helpful in studying Special as well as general theory of relativity.
• Dynamical systems would bring the concept of stability of a physical system, linear and non-linear aspects of a system.

Module - I

Constraints and Constrained Motion:

Constraints and their classification. Generalized coordinates and Degrees of freedom. Virtual displacement and virtual work, D’Alembert’s principle and its application. (3 Lectures)

Small oscillations:

Stable and unstable equilibriums. Small oscillations in a system with one degree of freedom. Small oscillations using generalized coordinates. Normal coordinates and frequencies of vibration. (3 Lectures)

Module – II

Lagrangian Formulation:

Variation techniques. Hamilton’s variational principle. Lagrange’s equations of motion from Hamilton’s principle(for conservative systems) and applications of Lagrange’s equations. Lagrange’s undetermined multipliers, Lagrange’s equation for non-holonomic systems, Cyclic coordinates. Invariance and Noether’s theorem. (6 Lectures)

Hamiltonian Formulation:

Hamilton’s function and Hamilton’s equations of motion. Phase space and the motion of the system. Applications of Hamiltonian formalism to simple problems. (4 Lectures)

Module – III

Canonical Transformations:

Canonical transformations, Conditions for transformation to be canonical. Poisson brackets, Lagrange and Poisson brackets as canonical invariants. Equations of motion in Poisson bracket notation. Jacobi identity. Infinitesimal contact transformation. Constants of the motion. Symmetry properties. Poisson bracket relations. The angular momentum and Poisson brackets. Liouville’s theorem. (8 Lectures)

Hamilton-Jacobi Theory:

The Hamilton Jacobi equation for Hamilton's principle function. The harmonic oscillator problem. Hamilton-Jacobi Equation for Hamilton’s characteristic function. (4 Lectures)

Module – IV

Rigid Body Dynamics:

Generalized coordinates for rigid body motion, Orthogonal transformations and rotations (finite and infinitesimal). Euler’s theorem, Euler angles. Angular velocity and angular momentum of rigid body, Inertia tensor and principal axis system. Euler’s equations. Heavy symmetrical top with precession and nutation. (7 Lectures)

Module – V

Special Theory of Relativity:

Michelson–Morley experiment and its outcome. Postulates of the Special Theory of Relativity. Lorentz transformations and their consequences, including simultaneity and order of events, length contraction, and time dilation. Relativistic addition of velocities. Variation of mass with velocity, concept of zero rest mass of photon, and mass–energy equivalence. Relativistic Doppler effect. Relativistic kinematics and transformation of energy and momentum.

Introduction to Minkowski four-dimensional space–time, space–time interval, time-like and space-like separations, four-vectors and their Lorentz transformation, with applications to relativistic kinematics and basic relativistic dynamics. (10 Lectures)

Tutorials (15 Hours)

The tutorial sessions are designed to support and reinforce the topics discussed in the lecture classes. These sessions will focus on solving problems, working through important derivations, and discussing key concepts in greater detail. Students will be encouraged to solve analytical and numerical problems, clarify doubts, and actively participate in discussions and presentations of assigned exercises.

These instructions will be followed for the tutorials of other papers/courses for Semesters I to IV.

References

1. Classical Mechanics, H. Goldstein, C.P. Poole, and J.L. Safko, Pearson, 3^rd Edition
2. Classical Mechanics, J. C. Upadhyaya, Himalaya Publishing House, 2^nd Edition.

Additional References

1. Classical Mechanics: The Theoretical Minimum, L Susskind and G Hrabovsky, Penguin
2. Classical Mechanics, N. C. Rana and P. S. Joag, Tata McGraw Hall.
3. Theoretical Mechanics, M. R. Spiegel, Schaum’s Outline Series, McGraw-Hill.
4. Fundamentals of Classical Mechanics, A.B. Gupta, Books and Allied Pvt Ltd.
5. Classical Mechanics, S. Chandra, Narosa Publishing House.