105f94-lectures

105f94-lectures - 1 SHORT COURSE IN CLASSICAL MECHANICS M....

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1 SHORT COURSE IN CLASSICAL MECHANICS M. Strovink University of California, Berkeley January 2, 2001 1. Vectors and Transformations. 1.1. Body and space axes. In ordinary 3-dimensional space, we require six independent quantities to specify the conFg- uration of a rigid body. (Take r 1 , r 2 ,and r 3 to be vectors from the origin to each of three ref- erence points in the body. If the body is rigid, | r 1 r 2 | , | r 2 r 3 | | r 3 r 1 | are Fxed, so the number of independent quantities is only six.) Three of these six may be identiFed with a vec- tor R from the origin to some basic reference point, e.g. the center of mass. The remaining three quantities are orientation variables . To study these orientation variables, we use a set of unprimed “body axes” ( x 1 ,x 2 3 )a t - tached to the rigid body, and a set of primed “space axes” ( x 0 1 0 2 0 3 ) using the same origin as the body axes, but having their directions Fxed to be the same as those of a particular set of external reference axes. The external frame is usually taken to be inertial, i.e. not accelerating. A point on the rigid body can be represented in either coordinate system. Going from one representation to another requires a linear transformation : x 0 1 = λ 11 x 1 + λ 12 x 2 + λ 13 x 3 x 0 2 = λ 21 x 1 + λ 22 x 2 + λ 23 x 3 x 0 3 = λ 31 x 1 + λ 32 x 2 + λ 33 x 3 (1 . 1) (We require the transformation to be linear so that it does not depend on the dimensions of x and x 0 .) Other notation for (1.1) is: x 0 i = 3 X j =1 λ ij x j x 0 i = λ ij x j where in the last expression summation from 1 to 3 over the repeated index j is assumed by con- vention . In matrix notation, we could also write ˜ x 0 =Λ˜ x where ˜ x = x 1 x 2 x 3 ˜ x 0 = x 0 1 x 0 2 x 0 3 are column vectors, and Λ= λ 11 λ 12 λ 13 λ 21 λ 22 λ 23 λ 31 λ 32 λ 33 isa3 × 3 matrix. Of course, the rules of matrix multiplication are followed. To test your understanding of matrix mul- tiplication and the convention that repeated in- dices are summed, consider the product C = AB where A , B C are 3 × 3 matrices. Then the ij element of C is given by C ij = A ik B kj . 1.2. Properties of the transformation matrix. The nine matrix elements λ 11 ...λ 33 depend on only three (as yet unspeciFed) orientation
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2 variables. Therefore, there must be six equa- tions that relate the matrix elements to each other (more on that below). The physical signif- icance of the λ ij is revealed by transforming the unit vectors ˆ e 1 = 1 0 0 ˆ e 2 = 0 1 0 ˆ e 3 = 0 0 1 into the primed system, as usual by multiplying them by Λ: Λˆ e 1 = λ 11 λ 21 λ 31 . Therefore, λ 11 is the projection of ˆ e 1 on the x 0 1 axis, λ 21 is the projection of ˆ e 1 on the x 0 2 axis, etc. Using the well-known property of the dot product cos θ ab = a · b | a || b | , we obtain λ 11 e 0 1 · ˆ e 1 =cos θ 1 0 1 λ 21 e 0 2 · ˆ e 1 θ 2 0 1 λ 31 e 0 3 · ˆ e 1 θ 3 0 1 , etc. That is, the λ ij are the direction cosines relating axis i 0 to axis j .
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105f94-lectures - 1 SHORT COURSE IN CLASSICAL MECHANICS M....

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