CLASSICAL MECHANICS

Basic concepts of coordinate systems and degrees of freedom; Holonomic and non-holonomic constraints; D-Alembert's Principle; Generalized coordinates; Principle of virtual work;  Lagrangian, Lagrange's equation and its applications; Velocity dependent potential in Lagrangian formulation; Generalized momentum; Hamilton's Principle; Lagrange’s equation from Hamilton's Principle.

 

Calculus of variation and its application to simple problems: Hamilton's Equations; Applications of Hamilton’s equation; Hamilton’s canonical equation in different coordinate systems; Calculus of Variation and Euler-Lagrange’s equations; Brachistochrone Problem; Derivation of Lagrange's and Hamilton’s canonical equation from Hamilton’s variational principle; Method of Lagrange's undetermined multipliers; Principle  of least action.

 

Conservation principle and Noether's theorem; Conservation of energy, linear momentum and angular momentum as a consequence of homogeneity of time and space and isotropy of space respectively.

 

Eulerian angles; Euler’s theorem; Angular momentum and Inertia Tensor; Euler’s equations; Euler’s equation of motion for a rigid body.

 

Canonical transformation: Legendre transformations; Generating functions; Conditions for Canonical transformation; Bilinear invariant conditions; Lagrange's and Poisson’s brackets and their relations; Angular momentum and Poisson Brackets; equation of motion in Poisson bracket formulation; Invariance of Poisson’s and Lagrange’s Brackets under canonical transformations; Liouville's theorem.                                                               

 

Action angle variable: Hamilton Jacobi equation and its applications; adiabatic invariance of action variable; The Kepler problem in action angle variables; theory of small oscillations in Lagrangian formulation; theory of small oscillations in normal coordinates and normal modes; Two coupled oscillator and solution of its differential equation; Two coupled pendulum; Double pendulum.