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The final publication is available at Springer via http://dx.doi.org/10.1007/s10569-017-9771-yWe outline some aspects of the dynamics of an infinitesimal mass under the Newtonian attraction of three point masses in a symmetric collinear relative equilibria configuration when a repulsive Manev potential (-1/r+e/r2-1/r+e/r2), e>0e>0, is applied to the central mass. We investigate the relative equilibria of the infinitesimal mass and their linear stability as a function of the mass parameter ßß, the ratio of mass of the central body to the mass of one of two remaining bodies, and e. We also prove the nonexistence of binary collisions between the central body and the infinitesimal mass.Peer ReviewedPostprint (published version

The spatial equilateral restricted four-body problem (ERFBP) is a four body problem where a mass point of negligible mass is moving under the Newtonian gravitational attraction of three positive masses (called the primaries) which move on circular periodic orbits around their center of mass fixed at the origin of the coordinate system such that their configuration is always an equilateral triangle. Since fourth mass is small, it does not affect the motion of the three primaries. In our model we assume that the two masses of the primariesm2andm3are equal toμand the massm1is1−2μ. The Hamiltonian function that governs the motion of the fourth mass is derived and it has three degrees of freedom depending periodically on time. Using a synodical system, we fixed the primaries in order to eliminate the time dependence. Similarly to the circular restricted three-body problem, we obtain a first integral of motion. With the help of the Hamiltonian structure, we characterize the region of the possible motions and the surface of fixed level in the spatial as well as in the planar case. Among other things, we verify that the number of equilibrium solutions depends upon the masses, also we show the existence of periodic solutions by different methods in the planar case.

We formulate the problem of an infinitesimal particle moving in the space under the influence of the gravitational force induced by a homogeneous annulus disk fixed on a plane. We compute using different coordinates and in terms of an elliptic integral the potential associated to this problem. Also we study the symmetries of the associated potential. After that, we look at the dynamics in some particular cases, namely: in the line perpendicular to the plane that contains the annulus disk and passes through the center of the ring and on the plane that contains the massive annulus.

BACKGROUND Transpetrosal approaches have been used for treatment of tumors in the petroclival region for many years. Injury to the temporal lobe, however, has been a potential drawback of the techniques described to date. OBJECTIVE To describe modifications of the transpetrosal surgical technique, which allows extradural manipulation of the temporal lobe during the focused combined transpetrosal approach. This extra layer of protection avoids mechanical brain retraction, direct trauma to the temporal lobe and disruption of the local venous structures. METHODS The present manuscript describes an innovative technical nuance based on the combination of the focused combined transpetrosal approach, the peeling of the dural layers of the tentorium, and the reverse peeling of the middle fossa dura mater. Ample illustrative material is provided and illustrative cases are presented. CONCLUSION Peeling of the dural layers of the tentorium is a promising modification of the transpetrosal approach to increase the safety of the temporal lobe manipulation.