1.5 KiB
PROBLEMS PROBLEM 1. Obtain expressions for the Cartesian components and the magnitude of the angular momentum of a particle in cylindrical co-ordinates r, , Z. SOLUTION. Mx = m(rz-zi) sin - -mrzo cos , My = -m(rz-zi) cos -mrzo sin , Mz = mr2 M2 = PROBLEM 2. The same as Problem 1, but in spherical co-ordinates r, 0, o. SOLUTION. Mx = -mr2(8 sin + sin 0 cos 0 cos ), My = mr2(j cos - sin 0 cos 0 sin b), Mz = mr2sin20, M2 = PROBLEM 3. Which components of momentum P and angular momentum M are conserved in motion in the following fields? (a) the field of an infinite homogeneous plane, (b) that of an infinite homogeneous cylinder, (c) that of an infinite homogeneous prism, (d) that of two points, (e) that of an infinite homo- geneous half-plane, (f) that of a homogeneous cone, (g) that of a homogeneous circular torus, (h) that of an infinite homogeneous cylindrical helix.
SOLUTION. (a) Px, Py, Mz (if the plane is the xy-plane), (b) M, Pz (if the axis of the cylinder is the z-axis), (c) P (if the edges of the prism are parallel to the z-axis), (d) Mz (if the line joining the points is the z-axis), (e) Py (if the edge of the half- plane is the y-axis), (f) Mz (if the axis of the cone is the z-axis), (g) Mz (if the axis of the torus is the z-axis), (h) the Lagrangian is unchanged by a rotation through an angle so about the axis of the helix (let this be the z-axis) together with a translation through a distance h86/2m along the axis (h being the pitch of the helix). Hence SL = 8z aL/dz+ +80 0L/26 = = 0, so that I+hPz/2n = constant.