Welcome to Classical Mechanics (PHYS 705) course page

HWs

1. Review of Mechanics: 

   • Show that the kinetic energy is given by Eqn 1 in plane polar coordinates.

   • Chpt 1, problems 1, 2, and 3.

   • [22 Aug; due: 29 Aug ]

2. Lagrange equations - I

   • Chpt 1, problems 7, 8, 9, and 10

   • [29 Aug; due: 12 Sept ]

3. Lagrange equations - II & Hamilton's principle

   • Chpt 1, problems 20, 22, 23, and Chpt 2, problem 1

   • [12 Sept; due: 19 Sept ]

4. Hamilton's principle & Calculus of variations

   • A geodesic is a line that represents the shortest path between any two points when the path is restricted to a particular surface. Show that the geodesics of a spherical surface are great circles, i.e., circles whose centers lie at the center of the sphere. 

   • Chpt 2, problems 12 and 14

   •  [19 Sept; due: 26 Sept ]

5. Central force problem

   • Chpt 2, problems 20, 21

   •  [26 Sept; due: 3 Oct ]

6. Kepler problem 

   • Chpt 3, problem 13 

   • [3 Oct; due: 11 Oct ]

7. Hamilton's equations

   • Chpt 8, problems 1,  16

   • A pendulum consists of a mass m suspended by a massless spring with unextended length b and spring constant k. 

(a) Sketch the problem.

(b)  Use the Lagrangian method to find the equation of motion.

(c) Determine the Hamiltonian and Hamilton's equation of motion.

7. • [11 Oct; due: 31 Oct ]

8. Canonical transformations

   • Chpt 9, problem 8

   • [31 Oct; due: 7 Nov ]

9. Hamilton- Jacobi Equation & Hamilton Principal Function

   • Chpt 10, problem 5

   • [7 Nov; due: 14 Nov]

10. Oscillations

    • Obtain the normal modes of vibration for the double pendulum shown in Figure 1.4 in Goldstein, assuming equal lengths, but not equal masses. Show that when the lower mass is small compared to the upper one, the two resonant frequencies are almost equal. If the pendula are set in motion by pulling the upper mass slightly away from the vertical and then releasing it, show that subsequent motion is such that at regular intervals one pendulum is at rest while the other has its maximum amplitude. This is the familiar phenomenon of beats. 

    • [14 Nov; due:  28 Nov ]