M.Sc., Physics. First Semester Title: Code No: PHY 101 Hard Core Paper Credit -3

Mechanics of a system of particles: Conservation of linear and angular momenta in the absence of (net) external and using centre of . The equation and the total of a system of particles using scalar potential.

The Lagrangean method: Constraints and their classifications. Generalized coordinates. Virtual , D’Alembert’s principle and Lagrangean equations of the second kind. Examples of (I) single particle in Cartesian, spherical polar and cylindrical polar coordinate systems, (II) Atwood’s machine and (III) a bead sliding on a rotating wire in a -free , (IV) Simple . Derivation of Lagrange equation from Hamilton principle.

Motion of a particle in a field: Binet equation for central (Lagrangean method).

[16 hours]

Hamilton’s equations: Generalized momenta. Hamilton’s equations. Examples—simple har- monic oscillator, simple pendulum, compound pendulum, of a particle in a central force field, charged particle moving in an electromagnetic field and Hamiltonian for a free particle in dif- ferent coordinates. Cyclic coordinates. Physical significance of the Hamiltonian function. Derivation of Hamilton’s equations from a variational principle.

Canonical transformations: Definition, Generating functions (Four basic types), examples of Canonical transformations. as an example to canonical transformation, In- finetisimal contact transformation. Poisson brackets; properties of Poisson brackets, angular mo- mentum and Poisson bracket relations. Equation of motion in the Poisson bracket notation. The Hamilton- Jacobi equation; the example of the harmonic oscillator treated by the Hamilton-Jacobi method.

[16 hours]

Mechanics of rigid bodies: Degrees of freedom of a free , Angular and of rigid body. of tensor, principal moments of inertia, products of inertia, the inertia tensor. Euler for a rigid body. free motion of a rigid body. Precession of earth’s axis of rotation, Euler angles, angular of a rigid body.

Small oscillations of mechanical system: Introduction, types of equilibria, Quadratic forms of kinetic and potential of a system in a equilibrium, General theory of small oscillations, secular equation and eigenvalue equation, small oscillations in normal coordinates and normal modes, examples of two coupled oscillators, of a linear triatomic molecule.

[16 hours]

References

• Upadhyaya J.C., Classical mechanics, Himalaya Publishing House, Mumbai. 2006. (Unit 1. Chapter 1 and 2 page No. 1-50.) • Goldstein H., Poole C. and Safko J., Classical mechanics, 3rd Edn., Pearson Education, New Delhi. 2002. (Unit 1, Chapter 1 page 1-14, 16-21, 24-29) • Srinivasa Rao K.N., Classical mechanics, Universities Press, Hyderabad. 2003. • Takwale R.G. and Puranik S., Introduction to classical mechanics, Tata McGraw, New Delhi,1991. • Landau L.D. and Lifshitz E.M., Classical mechanics, 4th Edn., Pergamon Press, 1985. Department of Physics, SPC, Mysore.