Diacu, Florin N.2010-01-082010-01-0819912010-01-08http://hdl.handle.net/1828/2055We first show that trapezoidal and rhomboidal solutions of the four-body problem with equal masses, do not lead to non-collision singularities and that any orbit encounters a collision, forwards or backwards in time. The existence of square homothetic solutions on negative energy levels, as connecting orbits between equilibria, is further proved and a transversality theorem, interpreted as a structure stability result, follows. Using McGehee transformations the total collapse singularity is blown up and the binary collisions are regularized. Finally we discuss the block-regularization of quadruple collision orbits as it follows from the qualitative behavior of the flow on the collision manifold.entechnical reports (mathematics and statistics)Technical ReportDepartment of Mathematics and Statistics