Welcome to Classical Mechanics (PHYS 705) course page
Fall 2022, George Mason University
[Aug 22, 2022 - Dec 14, 2022]
Fall 2022, George Mason University
[Aug 22, 2022 - Dec 14, 2022]
Exploratory Hall 1004 (See in map)
Monday at 16:30-19:10
Prof. Dr. Erdal Yiğit
Textbooks: (1) Classical Mechanics by Goldstein, Poole, Safko, 3rd edition. (2) Analytical Mechanics, Hand and Finch, Cambridge University Press.
Homework assignments are based on the problems in the book by Goldstein et al., unless otherwise stated. Chapter numbers refer to the chapters in Goldstein et al.
HWs
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 ]
Lagrange equations - I
Chpt 1, problems 7, 8, 9, and 10
[29 Aug; due: 12 Sept ]
Lagrange equations - II & Hamilton's principle
Chpt 1, problems 20, 22, 23, and Chpt 2, problem 1
[12 Sept; due: 19 Sept ]
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 ]
Central force problem
Chpt 2, problems 20, 21
[26 Sept; due: 3 Oct ]
Kepler problem
Chpt 3, problem 13
[3 Oct; due: 11 Oct ]
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.
[11 Oct; due: 31 Oct ]
Canonical transformations
Chpt 9, problem 8
[31 Oct; due: 7 Nov ]
Hamilton- Jacobi Equation & Hamilton Principal Function
Chpt 10, problem 5
[7 Nov; due: 14 Nov]
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 ]
HW5:
Problem 2.20: Repeat the calculations for the equations of motion in both coordinate systems
Problem 2.21: Assume that both springs have the same k. For the two set of generalized coordinates, one can take the standard Cartesian (x,y) coordinates as the “laboratory” generalized coordinates and in the rotating frame, one can use (r, l) as the generalized coordinates where (r + r0) is the location of the carriage on the main rotating beam and l is the location of the mass m along the second ridig beam attached perpendicular to the main beam.
HW1: Chpt 1, problem 3