Abstract
There exist two types of semi-direct products between a Lie group G and a vector space V. The left semi-direct product, G⋉ V, can be constructed when G is equipped with a left action on V. Similarly, the right semi-direct product, G⋊ V, can be constructed when G is equipped with a right action on V. In this paper, we will construct a new type of semi-direct product, \(G \bowtie V\), which can be seen as the ‘sum’ of a right and left semi-direct product. We then parallel existing semi-direct product Euler-Poincaré theory. We find that the group multiplication, the Lie bracket, and the diamond operator can each be seen as a sum of the associated concepts in right and left semi-direct product theory. Finally, we conclude with a toy example and the group of 2-jets of diffeomorphisms above a fixed point. This final example has potential use in the creation of particle methods for problems on diffeomorphism groups.
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Notes
- 1.
This observation was pointed out to us by Peter Michor.
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Acknowledgements
We would like to thank Darryl D. Holm for providing the initial stimulus for this project. The work of L. C has been supported by MICINN (Spain) Grant MTM2010-21186-C02-01, MTM 2011-15725-E, ICMAT Severo Ochoa Project SEV-2011-0087 and IRSES-project “Geomech-246981”. L. C owes additional thanks to CSIC and the JAE program for a JAE-Pre grant. The work of H.O. J was supported by European Research Council Advanced Grant 267382 FCCA.
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Colombo, L., Jacobs, H.O. (2015). Lagrangian Mechanics on Centered Semi-direct Products. In: Chang, D., Holm, D., Patrick, G., Ratiu, T. (eds) Geometry, Mechanics, and Dynamics. Fields Institute Communications, vol 73. Springer, New York, NY. https://doi.org/10.1007/978-1-4939-2441-7_9
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