On the stability of the Lagrangian homographic solutions in a curved three-body problem on $\mathbb{S}^2$
Regina Martínez Carles Simó
Discrete & Continuous Dynamical Systems - A 2013, 33(3): 1157-1175 doi: 10.3934/dcds.2013.33.1157
The problem of three bodies with equal masses in $\mathbb{S}^2$ is known to have Lagrangian homographic orbits. We study the linear stability and also a "practical'' (or effective) stability of these orbits on the unit sphere.
keywords: practical stability. Curved 3-body problem homographic orbits stability of solutions
Non-integrability of the degenerate cases of the Swinging Atwood's Machine using higher order variational equations
Regina Martínez Carles Simó
Discrete & Continuous Dynamical Systems - A 2011, 29(1): 1-24 doi: 10.3934/dcds.2011.29.1
Non-integrability criteria, based on differential Galois theory and requiring the use of higher order variational equations (VEk), are applied to prove the non-integrability of the Swinging Atwood's Machine for values of the parameter which can not be decided using first order variational equations (VE1).
keywords: differential Galois theory Non-integrability criteria Swinging Atwood Machine higher order variationals.
On the existence of doubly symmetric "Schubart-like" periodic orbits
Regina Martínez
Discrete & Continuous Dynamical Systems - B 2012, 17(3): 943-975 doi: 10.3934/dcdsb.2012.17.943
We give sufficient conditions to ensure the existence of symmetrical periodic orbits for a class of Hamiltonian systems having some singularities. The results are applied to different subproblems of the gravitational $n$-body problem where singularities appear due to collisions.
keywords: $n$-body problem collisions qualitative methods. Symmetric periodic orbits

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