Commutation Coef cients and Basis One Forms in .NET framework

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Commutation Coef cients and Basis One Forms
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It is also possible to determine whether or not a basis is holonomic by examining the basis one forms. A one form can be expanded in terms of a set of coordinate basis one forms a as = a a = a dx a In the same way that we can expand a vector in a different basis, we can also expand a one form in terms of a nonholonomic basis. Again using hats to denote the fact that we are working with a nonholonomic basis, we can write
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= a a
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Cartan s Structure Equations
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Both expansions represent the same one form. Given a particular set of basis one forms, it may be desirable to determine if it is holonomic. Once again we do this by calculating the commutation coef cients, but by a different method. Given a set of basis one forms a , the commutation coef cients can be found by calculating d a . This quantity is related to commutation coef cients in the following way: 1 d a = Cbc a b c 2 Now recall that for a coordinate basis, the basis one forms are given by a = dx a In the previous chapter, we learned that the antisymmetry of the wedge product leads to the following result for an arbitrary p form : d (d ) = 0 This means that for a coordinate basis, d a = 0. For spherical polar coordinates, if we choose the nonholonomic basis, the basis one forms are given by r = dr, = r d ,
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(5.6)
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= r sin d
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(5.7)
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Using (5.6), we can compute the commutation coef cients for this basis. For example, d = d (r sin d ) = sin dr d + r cos d d
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Using the de nitions given in (5.7), we can rewrite this expression in terms of the basis one forms. First, notice that 1 1 sin dr d = dr sin d = dr r sin d = r r r
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For the second term, we nd r cos d d =
Cartan s Structure Equations
cos cos (r d ) r d = (r d ) r sin d r r sin cot = r
Putting these results together, we obtain 1 cot d = r + r r The antisymmetry of the wedge product means we can write this expression as 1 cot d = r + r r 1 1 1 r 1 r + = 2 r r 2
cot cot r r
Now we compare with (5.6) to read off the commutation coef cients. We nd C r = C r =
and C = C =
cot r
Remember, if the commutation coef cients vanish, then the basis is holonomic. As we mentioned at the beginning of the chapter, it is often convenient to do calculations using the orthonormal basis but we may need to express results in the coordinate basis. We now explore the techniques used to transform between the two.
Transforming between Bases
We can work out a transformation law between the coordinate and noncoordinate basis vectors by using the coordinate components of the noncoordinate basis vectors. These components are denoted by (ea )b and known as the tetrad. The meaning of these components is the same as we would nd for any vector. In other words, we use them to expand a noncoordinate basis vector in terms of the basis vectors of a coordinate basis: ea = (ea )b eb (5.8)
Cartan s Structure Equations
For example, we can expand the noncoordinate basis for spherical polar coordinates in terms of the coordinate basis vectors as follows: er = (er )b eb = (er )r er + (er ) e + (er ) e e = e e = e
eb = e eb = e
er + e
e + e
er + e
e + e
Earlier we stated that the noncoordinate basis vectors were er = r , 1 e = , r e = 1 r sin
Comparison with the above indicates that (er )r = 1 1 e = r 1 e = r sin All other components are zero. The components (ea )b can be used to construct a transformation matrix that we label a b as this matrix represents a transforma tion between the global coordinates and the local Lorentz frame of an observer. In the case of spherical polar coordinates, we have 1 0 0 1 = r 0 0 0 0
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