Guide to Geometric Algebra in Practice pp 371-389 | Cite as

# Rigid Body Dynamics in a Constant Curvature Space and the ‘1D-up’ Approach to Conformal Geometric Algebra

## 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.

## Keywords

Angular Momentum Rigid Body Curvature Scale Curve Space Euclidean Geometry## References

- 1.Doran, C.J.L., Lasenby, A.N., Challinor, A.D., Gull, S.F.: Effects of spin-torsion in gauge theory gravity. J. Math. Phys.
**39**(6), 3303–3321 (1998) MathSciNetMATHCrossRefGoogle Scholar - 2.Heath, R.S.: On the dynamics of a rigid body in elliptic space. Philos. Trans. R. Soc. Lond.
**175**, 281–324 (1884) MATHCrossRefGoogle Scholar - 3.Lasenby, A.N.: Conformal geometry and the Universe. Unpublished paper available on the site http://www.mrao.cam.ac.uk/~clifford/publications (2003)
- 4.Lasenby, A.N.: Recent applications of conformal geometric algebra. In: Li, H., Olver, P.J., Sommer, G. (eds.) Computer Algebra and Geometric Algebra with Applications. Lecture Notes in Computer Science, p. 298. Springer, Berlin (2005) CrossRefGoogle Scholar
- 5.Lasenby, A.N.: Some results in the conformal geometry approach to the Dirac equation, electromagnetism and gravity (2011, in preparation) Google Scholar
- 6.Papapetrou, A.: Spinning test-particles in general relativity. I. Philos. Trans. R. Soc. Lond. A
**209**, 248–258 (1951) MathSciNetMATHGoogle Scholar