Stability of the rhomboidal symmetricmass orbit
Lennard Bakker  275 TMCB, Brigham Young University, Provo, UT 84602, United States (email) Abstract: We study the rhomboidal symmetricmass $1$, $m$, $1$, $m$ fourbody problem in the fourdegreesoffreedom setting, where $0 < m \leq 1$. Under suitable changes of variables, isolated binary collisions at the origin are regularizable. Analytic existence of the orbit in the fourdegreesoffreedom setting is established. We analytically extend a method of Roberts to perform linear stability analysis in this setting. Linear stability is analytically reduced to computing three entries of a $4 \times 4$ matrix related to the monodromy matrix. Additionally, it is shown that the fourdegreesoffreedom setting has a twodegreesoffreedom invariant set, and linear stability results in the subset comes ``for free'' from the calculation in the full space. The final numerical analysis shows that the fourdegreesoffreedom orbit is linearly unstable except for a very small interval about $m = 0.4$, whereas the twodegreesoffreedom orbit is linearly stable for all but very small values of $m$.
Keywords: nbody problem, binary collision, regularization, linear stability, rhomboidal problem.
Received: August 2013; Revised: June 2014; Available Online: August 2014. 
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