MSc Physics (Electronics I) SEM I 2023 2024 Feb 2024 CLASSICAL MECHANICS Question Paper - Mumbai University | munotes

Questions asked in this paper

• (2) Figures to the right indicate full marks

1. Q1 (a) Attempt any one:
   • (i) | What is the D Alembert’s Principle? Derive Lagrange’s equation from 7
   • (ii) State Hamilton’s principle. Derive Lagrangian equation from Hamilton’s 7
   • (b) Attempt any one:
   • (i) is velocity dependent potential? Give an example 3
   • (ii) By Lagrangian mechanics derive the equation of motion of a single particle 3 in 3 D space using Cartesian coordinates and show that they are equivalent to Newton’s laws
2. Q2 (a) Attempt any one:
   • (i) Show that isotropy of space leads to conservation of Angular momentum. 7
   • (ii) Obtain the expression for angular momentum and total energy (first 7 integrals) for motion under central force
   • (b) Attempt any one:
   • (i) State Kepler’s laws of planetary motion 3
   • (ii) The maximum and minimum velocities of a satellite are and v2 3 respectively, find the eccentricity of the orbit of the satellite
3. Q3 (a) Attempt any one:
   • (i) particle near the minima of the potential function, show that 7 Where symbols have their usual meanings
   • (ii) What are Legendre transformations? Derive Hamilton’s equations of motion 7
   • (b) Attempt any one: What are three types of equilibrium? 3
   • (ii) What is cyclic coordinate? Explain with an example. 3
4. Q4 (a) Attempt any one:
   • (i) are generating functions for canonical transformations. Explain the 7 four types of generating functions
   • (ii) What are Poisson’s brackets? Show that they remain invariant under 7
   • (b) Attempt any one:
   • (i) Obtain the equation of motion in Poisson bracket form. 3
   • (ii) Show that the following transformation is canonical 3
5. Q5 Attempt any five:
   • (a) Obtain the degrees of freedom of a simple pendulum. 2
   • (b) Define Holonomic and Non-Holonomic constraints 2
   • (c) State Virial theorem. 2
   • (d) Explain the terms impact parameter and differential scattering cross section. 2
   • (e) Consider the following Lagrangian: 2 Which coordinate is cyclic and why?
   • (f) State and explain variational principle. 2
   • (g) Verify whether the transformation Q = = and P = is canonical. 2
   • (h) Explain an exact differential condition for a transformation to be canonical. 2

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