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Aerobatics of Flying Saucers

Abstract

Starting from the observation that a flying saucer is a nonholonomic mechanical system whose 5-dimensional configuration space is a contact manifold, we show how to enrich this space with a number of geometric structures by imposing further nonlinear restrictions on the saucer’s velocity. These restrictions define certain ‘manœuvres’ of the saucer, which we call ‘attacking,’ ‘landing,’ or ‘\(G_2\) mode’ manœuvres, and which equip its configuration space with three kinds of flat parabolic geometry in five dimensions. The attacking manœuvre corresponds to the flat Legendrean contact structure, the landing manœuvre corresponds to the flat hypersurface type CR structure with Levi form of signature (1, 1), and the most complicated \(G_2\) manœuvre corresponds to the contact Engel structure (Engel in C R Acad Sci 116:786–788, 1893; Mano et al. in The geometry of marked contact twisted cubic structures, 2018, arXiv:1809.06455) with split real form of the exceptional Lie group \(G_2\) as its symmetries. A celebrated double fibration relating the two nonequivalent flat 5-dimensional parabolic \(G_2\) geometries is used to construct a ‘\(G_2\) joystick,’ consisting of two balls of radii in ratio \(1\!:\!3\) that transforms the difficult \(G_2\) manœuvre into the pilot’s action of rolling one of joystick’s balls on the other without slipping nor twisting.

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Notes

  1. In the sense that the common stabilizer of \({\Upsilon }\) and \({\text {d}}\omega ^0\) is irreducible \(\mathbf {GL}(2,\mathbb {R})\).

References

  1. Bryant, R.L.: Two exotic holonomies in dimension four, path geometries, and twistor theory. In: Complex Geometry and Lie Theory (Sundance, UT, 1989), Proceedings of Symposium in Pure Mathematics, American Mathematical Society, vol. 53, pp. 33–88 (1991)

  2. Bryant, R.L.: Élie Cartan and geometric duality. J. Élie Cartan 1998 1999 Inst. Cartan 16, 5–20 (2000)

    MATH  Google Scholar 

  3. Cartan, É.: Les systèmes de pfaff à cinq variables et les équations aux dérivées partielles du seconde ordre. Ann. Sci. Norm. Super. 27, 109–192 (1910)

    Article  Google Scholar 

  4. Čap, A., Slovák, J.: Parabolic Geometries I, Background and General Theory, Mathematical Surveys and Monographs, vol. 154. American Mathematical Society, Providence (2009)

    MATH  Google Scholar 

  5. Eastwood, M.G., Nurowski, P.: Aerodynamics of flying saucers. Commun. Math. Phys. (2020). https://doi.org/10.1007/s00220-019-03622-1

    Article  Google Scholar 

  6. Engel, F.: Sur un groupe simple a quatorze parametres. C. R. Acad. Sci. 116, 786–788 (1893)

    MATH  Google Scholar 

  7. Mano, G., Nurowski, P., Sagerschnig, K.: The geometry of marked contact twisted cubic structures. (2018). arXiv:1809.06455

  8. Nurowski, P.: Differential equations and conformal structures. J. Geom. Phys. 55, 19–49 (2005)

    ADS  MathSciNet  Article  Google Scholar 

  9. Nurowski, P.: Comment on \({{\mathbf{GL}}}(2,{\mathbb{R}})\) geometry of 4th order ODEs. J. Geom. Phys. 59, 267–278 (2009)

    ADS  MathSciNet  Article  Google Scholar 

  10. Nurowski, P.: On exceptional contact geometries, talk at the Australian National University, Canberra. (2013). http://www.fuw.edu.pl/~nurowski/prace/talk_canberra.pdf

  11. https://en.wikipedia.org/wiki/Flying_saucer

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Acknowledgements

The authors would like to thank Katja Sagerschnig and Travis Willse for many helpful conversations. Special thanks are due to Jan Gutt for the idea of a \(G_2\) joystick, which was suggested during a beer session with the second author at Jabeerwocky, one of the craft beer pubs in Warsaw.

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Correspondence to Paweł Nurowski.

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This work was supported by the Simons Foundation Grant 346300 and the Polish Government MNiSW 2015–2019 matching fund. It was written whilst the first author was visiting the Banach Centre at IMPAN in Warsaw for the Simons Semester ‘Symmetry and Geometric Structures’ and during another visit to Warsaw supported by the Polish National Science Centre (NCN) via the POLONEZ grant 2016/23/P/ST1/04148, which received funding from the European Union’s Horizon 2020 research and innovation programme under the Marie Skłodowska-Curie Grant agreement No. 665778

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Communicated by P. Chrusciel

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Eastwood, M., Nurowski, P. Aerobatics of Flying Saucers. Commun. Math. Phys. 375, 2335–2365 (2020). https://doi.org/10.1007/s00220-019-03621-2

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  • DOI: https://doi.org/10.1007/s00220-019-03621-2