hw2_problems

# hw2_problems - mation in terms of the direction cosines of...

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Homework 2 Classical Mechanics (Phys701) Due: September 19, 2011 Problem 4-2 Prove the following properties of the transposed and adjoined matrices ( A · B ) T = B T · A T , ( A · B ) = B · A Problem 4-3 Show that the trace of the matrix is invariant under any similarity transformation. Show also that antisymmetric property of a matrix is preserved under orthogonal similarity transformation. Problem 4-11 Verify the relationship det( - ˆ B ) = ( - 1) n det ˆ B for the determinant of an n × n matrix ˆ B . Problem 4-12 In a set of axes where z axis is the axis of a ﬁnite rotation, the rotation matrix is given by equation (4.43) R = cos φ sin φ 0 - sin φ cos φ 0 0 0 1 Derive the rotation formula (4.62) r 0 = r cos φ + n ( n · r )(1 - cos φ ) + [ r × n ]sin φ by transforming to an arbitrary coordinate system, expressing the orthogonal matrix of transfor-
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Unformatted text preview: mation in terms of the direction cosines of the axis of the nite rotation. Hint: express the second formula in the matrix form for a general direction of n given by the polar angles ( , ). Check that when n is along z , i.e. for = 0, you recover the matrix expression (4.43). Problem 4-13 (b) Show that a rotation about any given axis can be obtained as the product of two successive rotation, each through 180 . Problem 4-14 (A) Verify the the permutation symbol satises the following identity in terms of Kronecker delta symbols ijp rmp = ir jm- im jr . (B) Show that ijp ijk = 2 pk . 1...
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## This note was uploaded on 12/13/2011 for the course PHYS 701 taught by Professor Bazaliy during the Fall '11 term at South Carolina.

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