Abstract
We discuss a ‘1D up’ approach to Conformal Geometric Algebra, which treats the dynamics of rigid bodies in 3D spaces with constant curvature via a 4D conformal representation. All equations are derived covariantly from a 4D Lagrangian, and definitions of energy and momentum in the curved space are given. Some novel features of the dynamics of rigid bodies in these spaces are pointed out, including a simple non-relativistic version of the Papapetrou force in General Relativity. The final view of ordinary translational motion that emerges is perhaps surprising, in that it is shown to correspond to precession in the 1D up conformal space. We discuss the alternative approaches to Euclidean motions and rigid body dynamics outlined by Gunn in Chap. 15 and Mullineux and Simpson in Chap. 17 of this volume, which also use only one extra dimension, and compare these with the Euclidean space limit of the current approach.
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Notes
- 1.
Editorial note: The notation n and \(\bar{n}\) is used in [4]. When we compare to the notation used elsewhere in this volume we find that n=n ∞ and \(\bar {n}= -2n_{o}\). Correspondingly, \(e = \frac{1}{2}(n+\bar{n}) = -n_{o}+ \frac {1}{2} n_{\infty}\) and \(\bar{e}= \frac{1}{2}(n-\bar{n}) =n_{o}+ \frac{1}{2} n_{\infty}\). Note that \(\bar{n}\cdot n = 2\) corresponds to n o ⋅n ∞=−1. As a compromise between the notation in this book and [4] that avoids awkward factors, in this chapter we will use \(\bar{n}\), but replace n by n ∞.
- 2.
Those interested might like to know that if we extend this work to the space–time, rather than the purely spatial case, then the positive curvature version of the space we obtain is called de Sitter space, and λ is related to the usual cosmological constant, Λ (as measured in inverse metres squared), by Λ=12/λ 2. (See e.g. [3].)
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Lasenby, A. (2011). Rigid Body Dynamics in a Constant Curvature Space and the ‘1D-up’ Approach to Conformal Geometric Algebra. In: Dorst, L., Lasenby, J. (eds) Guide to Geometric Algebra in Practice. Springer, London. https://doi.org/10.1007/978-0-85729-811-9_18
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DOI: https://doi.org/10.1007/978-0-85729-811-9_18
Publisher Name: Springer, London
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