Reduction and relative equilibria for the 2-body problem in spaces of constant curvature

Borisov, A.V. and García-Naranjo, L.C. and Mamaev, I.S. and Montaldi, James (2017) Reduction and relative equilibria for the 2-body problem in spaces of constant curvature. [MIMS Preprint]

WarningThere is a more recent version of this item available.
[img] Text
curved2BP.pdf

Download (610kB)

Abstract

We perform the reduction of the two-body problem in the two dimensional spaces of constant non-zero curvature and we use the reduced equations of motion to classify all relative equilibria (RE) of the problem and to study their stability. In the negative curvature case, the nonlinear stability of the stable RE is established by using the reduced Hamiltonian as a Lyapunov function. Surprisingly, in the positive curvature case this approach is not sufficient and the proof of nonlinear stability requires the calculation of Birkhoff normal forms and the application of stability methods coming from KAM theory. In both cases we summarize our results in the Energy-Momentum bifurcation diagram of the system.

Item Type: MIMS Preprint
Subjects: MSC 2010, the AMS's Mathematics Subject Classification > 70 Mechanics of particles and systems
Depositing User: Dr James Montaldi
Date Deposited: 12 Aug 2017
Last Modified: 08 Nov 2017 18:18
URI: http://eprints.maths.manchester.ac.uk/id/eprint/2567

Available Versions of this Item

Actions (login required)

View Item View Item