Homework 9 - Physics 601 Homework 10   Due...

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Unformatted text preview: Physics 601 Homework 10   Due Friday November 12 Goldstein ­ ­ ­4.22, 4.24, 4.25 0 1. This problem concerns rotations about the z axis. ȹ cos(θ ) sin(θ ) 0ȹ ȹ ȹ a. Show that the rotation about the z axis: R = ȹ − sin(θ ) cos(θ ) 0ȹ can be ȹ ȹ 0 1Ⱥ ȹ 0 ȹ 0 −1 0ȹ ȹ ȹ written as R = exp( −θM z ) where M z = ȹ 1 0 0ȹ . ȹ ȹ € ȹ 0 0 0Ⱥ 1 b. Show that cos(ϑ ) = 1 tr ( R) − 2 2 2. There is € general theorem by Euler that any rotation matrix can be represented as a a rotation about one given axis. Thus by analogy to problem 3a. it can be written € ȹ n x ȹ € ȹ ȹ ˆ ˆ as R = exp −Φn ⋅ M = exp −Φ n x M x + n y M y + n z M z where n = ȹ n y ȹ is the unit ȹ ȹ ȹ n z Ⱥ vector specify the axis of rotation and Φ is the angle specifying the rotation. The purpose of this problem is to find the explicit of the rotation matrix for such a € ˆ rotation. As a first step note that n is completely specified by a polar angle θ and € azimuthal angle φ. Define Rnˆ ≡ Rz (φ ) Ry (θ ) . T ˆ a. As a first step show that n ⋅ M = Rnˆ M z Rnˆ . T € ˆ b. Show that R = exp( −Φ n ⋅ M ) = Rnˆ exp( − ΦM z ) Rnˆ € c. Express the nine matrix elements of R in terms of the polar and azimuthal angles defining n and the rotation angle Φ. €ˆ Not that this description of the rotation matrices is alternative parameterization to € the Euler angles. € € 3. For the general rotation of the form given in 2: ˆ a. Show that n is an eignevector of R with eigenvalue one. b. Show that cos(Φ) = 1 Tr ( R) − 1 . 2 2 4. Suppose we have a rotation specified by the Euler angles ( π , π , π ) (that is 333 πππ R = Rz ( 3€Rx ( 3 ) Rz ( 3 ) ) . Find the angle of rotation about the single axis and the axis ) € ˆ n . € € ( € € ) (( )) ...
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