By George W. Collins
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So important are these rotation matrices that it is worth remembering their form so that they need not be re-derived every time they are needed. One can show that it is possible to get from any given orthogonal coordinate system to another through a series of three successive coordinate rotations. Thus a general orthonormal transformation can always be written as the product of three coordinate rotations about the orthogonal axes of the coordinate systems. It is important to remember that the matrix product is not commutative so that the order of the rotations is important.
6) yield the equations of motion and usually one of the constants of the motion has been found in the process. This formalism is so powerful that it forms the basis for a great deal of quantum mechanics. It is clear that for celestial mechanics, the central remaining problem to finding the equations of motion is the determination of the potential and so in the next chapter we will turn to various methods by which that can be done. 49 Chapter 3: Exercises 1. The escape velocity from the Earth is the minimum velocity required to escape the influence of the Earth's gravitational field.
4) i This must be true for all vectors in the coordinate system so that ∑ A ijA ik = δ jk = ∑ A −ji1A ik i . 5) i Now remember that the Kronecker delta δ ij is the unit matrix and any element of a group that multiplies another and produces that group's unit element is defined as the inverse of that element. Therefore A ji = A ij −1 . 6) [ ] 22 Interchanging the elements of a matrix produces a new matrix which we have called the transpose of the matrix. Thus orthogonal transformations that preserve the length of vectors have inverses that are simply the transpose of the original matrix so that A −1 = A T .
