Blow up of the isosceles 3--body problem with an infinitesimal mass
Martha Alvarez-Ramírez Joaquín Delgado
Discrete & Continuous Dynamical Systems - A 2003, 9(5): 1149-1173 doi: 10.3934/dcds.2003.9.1149
We consider the isosceles $3$--body problem with the third particle having a small mass which eventually tend to zero. Classical McGehee's blow up is not defined because the matrix of masses becomes degenerate. Following Elbialy [3] we perform the blow up using the Euclidean norm in the planar $3$--body problem. We then complete the phase portrait of the flow in the collision manifold giving the behavior of some branches of saddle points missing in [3]. The homothetic orbits within the fixed energy level then provide the necessary recurrence in order to build a symbolic dynamics. This is done following ideas of S. Kaplan [6] for the collinear $3$--body problem. Here the difficulty is the larger number of critical points.
keywords: symbolic dynamics. blow up restricted isosceles problem 3–body problem
Non-integrability criterium for normal variational equations around an integrable subsystem and an example: The Wilberforce spring-pendulum
Primitivo B. Acosta-Humánez Martha Alvarez-Ramírez David Blázquez-Sanz Joaquín Delgado
Discrete & Continuous Dynamical Systems - A 2013, 33(3): 965-986 doi: 10.3934/dcds.2013.33.965
In this paper we analyze the non-integrability of the Wilbeforce spring-pendulum by means of Morales-Ramis theory in where is enough to prove that the Galois group of the variational equation is not virtually abelian. We obtain these non-integrability results due to the algebrization of the variational equation falls into a Heun differential equation with four singularities and then we apply Kovacic's algorithm to determine its non-integrability.
keywords: wilbeforce pendulum Algebrization hamiltonian systems differential Galois group. Kovacic's algorithm

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