Physics 422 -- Fall 2012

Homework #2, due Friday Sep. 14 at beginning of class

1.         [6pts] Marion & Thornton: problem 7.4:
(a) Find the Lagrange equations of motion for r and theta (polar coordinates). You do not need to solve these equations. But notice that the Lagrangian does not depend on theta, which gives a constant of the motion (the angular momentum).
 
(b) Use the two constants of the motion -- angular momentum and total energy -- to derive an equation for dr/dt as a function of r. (The spirit of this approach is that instead of solving the two Lagrange equations derived in part (a) directly, you are noticing that the Lagrangian does not explicitly depend on the time, which implies that the Hamiltonian -- in this case equal to T + V -- is a constant of the motion. You can therefore use the simpler equation T+V=constant to replace one of the two Lagrange equations derived in part (a).)
 
(c) Use that equation to write the solution to the problem in the form t-t0 = integral of a function of r. You do not need to carry out that integral.

2.         [6pts] Marion & Thornton: problem 7.7:
(a) Using the angles of the strings with respect to the vertical direction as the two generalized coordinates, find the equations of motion, without assuming the angles are small. (One way to compute the kinetic energy -- the only good way I know of -- is to find the locations of the masses in Cartesian coordinates, in terms of theta1 and theta2. You can then make good use of the trig identity cos(theta1 - theta2) = cos(theta1)*cos(theta2) + sin(theta1)*sin(theta2) .)
 
(b) Find the equations of motion in the limit of small oscillations. That limit is defined by keeping only the first-order terms in theta1 and theta2 in the equations of motion. In doing this, you can treat the time derivatives of theta1 or theta2 as being of the same order as those angles themselves. (Another way to obtain the equations of motion in the small-angle approximation limit would be to make appropriate small-angle approximations in the Lagrangian before finding the Lagrange equations of motions. You will learn how to do that, and how to solve the equations, later in this course.)