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The usual choice for a coordinate to supplement ξ in this case would be the polar angle. This coordinate is in agreement with Eq. (5.2), but does not obey the orthonormality condition of Eq. (5.1). It takes some straightforward manipulations to find the correct Jacobian for this choice. The method by Ruiz-Montero et al. was particularly designed to circumvent this need to specify the supplementary coordinates
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The usual choice for a coordinate to supplement ξ in this case would be the polar angle. This coordinate is in agreement with Eq. (5.2), but does not obey the orthonormality condition of Eq. (5.1). It takes some straightforward manipulations to find the correct Jacobian for this choice. The method by Ruiz-Montero et al. was particularly designed to circumvent this need to specify the supplementary coordinates.
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