N-body dynamics on closed surfaces : the axioms of mechanics
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A major challenge for our understanding of the mathematical basis of particle dynamics is the formulation of N-body and N-vortex dynamics on Riemann surfaces. In this paper, we show how the two problems are, in fact, closely related when considering the role played by the intrinsic geometry of the surface. This enables a straightforward deduction of the dynamics of point masses, using recently derived results for point vortices on general closed differentiable surfaces M endowed with a metric g. We find, generally, that Kepler's Laws do not hold. What is more, even Newton's First Law (the law of inertia) fails on closed surfaces with variable curvature (e.g. the ellipsoid).
Boatto , S , Dritschel , D G & Schaefer , R G 2016 , ' N-body dynamics on closed surfaces : the axioms of mechanics ' Proceedings of the Royal Society A - Mathematical, Physical & Engineering Sciences , vol 472 , no. 2192 , 20160020 . DOI: 10.1098/rspa.2016.0020
Proceedings of the Royal Society A - Mathematical, Physical & Engineering Sciences
© 2016, the Author(s). This work is made available online in accordance with the publisher’s policies. This is the author created, accepted version manuscript following peer review and may differ slightly from the final published version. The final published version of this work is available at rspa.royalsocietypublishing.org / https://dx.doi.org/10.1098/rspa.2016.0020
D.G.D. gratefully acknowledges support for this research from CNPq (Conselho Nacional de Desenvolvimento Cientifico e Tecnologico ) and FINEP (Inovação e Pesquisa) in Brazil, and from the UK Engineering and Physical Sciences Research Council (grant no. EP/H001794/1)
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