N-body dynamics on closed surfaces: the axioms of mechanics.
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 surfacesMendowed with a metricg. 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).
Departamento de Matemática Aplicada , Instituto de Matemática, Universidade Federal de Rio de Janeiro , Rio de Janeiro, CEP 21941-909, Brazil.
Stefanella
0000-0001-5708-0281
S
Boatto
School of Mathematics and Statistics , University of St Andrews , North Haugh, St Andrews KY16 9SS, UK.
David G
DG
Dritschel
Department de Matemàtica Aplicada , Facultat de Matemàtiques i Estadística, Universitat Politècnica de Catalunya Pau Gargallo , 5. Barcelona 08028, Spain.
Rodrigo G
RG
Schaefer
Hamiltonian systems; N-body problem; Newton's Laws; equivalence principle; point-vortex dynamics; surfaces of revolution