Regular Polygonal Equilibria on $$\mathbb {S}^1$$ and Stability of the Associated Relative Equilibria
Tóm tắt
For the curved n-body problem in
$$\mathbb {S}^3$$
, we show that a regular polygonal configuration for n masses on a geodesic is an equilibrium if and only if n is odd and the masses are equal. The equilibrium is associated with a one-parameter family (depending on the angular velocity) of relative equilibria, which take place on
$$\mathbb {S}^1$$
embedded in
$$\mathbb {S}^2$$
. We then study the stability of the associated relative equilibria on
$$\mathbb {S}^1$$
and
$$\mathbb {S}^2$$
. We show that they are Lyapunov stable on
$$\mathbb {S}^1$$
, they are Lyapunov stable on
$$\mathbb {S}^2$$
if the absolute value of angular velocity is larger than a certain value, and that they are linearly unstable on
$$\mathbb {S}^2$$
if the absolute value of angular velocity is smaller than that certain value.
Tài liệu tham khảo
Arnold, V.I., Kozlov, V.V., Neishtadt, A.I.: Mathematical Aspects of Classical and Celestial Mechanics (Dynamical Systems. III), Translated from the Russian original by E. Khukhro, 3rd edn., Encyclopaedia of Mathematical Sciences, Vol. 3. Springer, Berlin (2006)
Borisov, A.V., Mamaev, I.S., Kilin, A.A.: Two-body problem on a sphere, reduction, stochasticity, periodic orbits. Regul. Chaot. Dyn. 9(3), 265–279 (2004)
Cors, J.M., Roberts, G.E.: Four-body co-circular central configurations. Nonlinearity 25(2), 343–370 (2012)
Diacu, F., Pérez-Chavela, E.: Homographic solutions of the curved 3-body problem. J. Differ. Equ. 250(1), 340–366 (2011)
Diacu, F., Pérez-Chavela, E., Santoprete, M.: The \(n\)-body problem in spaces of constant curvature: part I, relative equilibria. J. Nonlinear Sci. 22(2), 247–266 (2012)
Diacu, F.: Relative equilibria of the 3-dimensional curved \(n\)-body problem. Mem. Am. Math. Soc. 228, 1071 (2013)
Diacu, F., Sánchez-Cerritos, J.M., Zhu, S.: Stability of fixed points and associated relative equilibria of the 3-body problem on \({\mathbb{S}}^1\) and \({\mathbb{S}}^2\). J. Dyn. Differ. Equ. 30(1), 209–225 (2018). Modification after publication at arXiv:1603.03339
Diacu, F., Stoica, C., Zhu, S.: Central configurations of the curved \(n\)-body problem. J. Nonlinear Sci. 28(5), 1999–2046 (2018)
Martínez, R., Simó, J.C.: On the stability of the Lagrangian homographic solutions in a curved three-body problem on \({\mathbb{S}}^2\). Discrete Contin. Dyn. Syst. 33(3), 1157–1175 (2013)
Meyer, K.R., Hall, G.R., Offin, D.: Introduction to Hamiltonian Dynamical Systems and the \(N\)-Body Problem. Applied Mathematical Sciences, 90, 2nd edn. Springer, New York (2009)
Moeckel, R.: Linear stability analysis of some symmetrical classes of relative equilibria. Hamilt. Dyn. Syst. 63, 291–317 (1992)
Patrick, G.: Relative equilibria in Hamiltonian systems: the dynamic interpretation of nonlinear stability on a reduced phase space. J. Geom. Phys. 9(2), 111–119 (1992)
Perko, L.M., Walter, E.L.: Regular polygon solutoins of the \(n\)-body problem. Proc. Am. Math. Soc. 94(2), 301–309 (1985)
Roberts, G.E.: Spectral instability of relative equilibria in the planar n-body problem. Nonlinearity 12(4), 757–769 (1999)
Stoica, C.: On the \(n\)-body problem on surfaces of revolution. J. Differ. Equ. 264(10), 6191–6225 (2018)