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2017.27: Reduction and relative equilibria for the 2-body problem in spaces of constant curvature

2017.27: A.V. Borisov, L.C. García-Naranjo, I.S. Mamaev and J. Montaldi (2017) Reduction and relative equilibria for the 2-body problem in spaces of constant curvature.

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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 2000 > 70 Mechanics of particles and systems
MIMS number:2017.27
Deposited By:Dr James Montaldi
Deposited On:12 August 2017

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