# A generalized Sitnikov problem

@article{Beltritti2017AGS, title={A generalized Sitnikov problem}, author={Gast{\'o}n Beltritti and Fernando Mazzone and Martina Oviedo}, journal={arXiv: Mathematical Physics}, year={2017} }

In this paper we address a $n+1$-body gravitational problem governed by the Newton's laws, where $n$ primary bodies orbit on a plane $\Pi$ and an additional massless particle moves on the perpendicular line to $\Pi$ passing through the center of mass of the primary bodies. We find a condition for that the configuration described be possible. In the case that the primaries are in a rigid motion we classify all the motions of the massless particle. We study the situation when the massless…

## 2 Citations

### Comet and Moon Solutions in the Time-Dependent Restricted $$(n+1)$$ ( n + 1 )

- Physics, GeologyJournal of Dynamics and Differential Equations
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The time-dependent restricted $$(n+1)$$ ( n + 1 ) -body problem concerns the study of a massless body (satellite) under the influence of the gravitational field generated by n primary bodies…

### Comet and Moon Solutions in the Time-Dependent Restricted ( n + 1 ) -Body Problem

- Physics, Geology
- 2021

The time-dependent restricted ( n + 1 ) -body problem concerns the study of a massless body (satellite) under the inﬂuence of the gravitational ﬁeld generated by n primary bodies following a periodic…

## References

SHOWING 1-10 OF 33 REFERENCES

### The Characterization of the Variational Minimizers for Spatial Restricted $N+1$-Body Problems

- Mathematics
- 2013

We use Jacobi's necessary condition for the variational minimizer to study the periodic solution for spatial restricted -body problems with a zero mass on the vertical axis of the plane for equal…

### On the classification of pyramidal central configurations

- Mathematics
- 1996

We present some results associated with the existence of central configurations (c.c.'s) in the classical gravitational N-body problem of Newton. We call a central configuration of five bodies, four…

### On the existence of collisionless equivariant minimizers for the classical n-body problem

- Mathematics, Physics
- 2004

We show that the minimization of the Lagrangian action functional on suitable classes of symmetric loops yields collisionless periodic orbits of the n-body problem, provided that some simple…

### The stability of vertical motion in the N-body circular Sitnikov problem

- Mathematics, Physics
- 2009

We present results about the stability of vertical motion and its bifurcations into families of 3-dimensional (3D) periodic orbits in the Sitnikov restricted N-body problem. In particular, we…

### Periodic Solutions in the Generalized Sitnikov (N+1)-Body Problem

- MathematicsSIAM J. Appl. Dyn. Syst.
- 2013

According to the number of bodies the authors prove the existence (or nonexistence) of a finite (or infinite) number of symmetric families of periodic solutions.

### Periodic orbits and bifurcations in the Sitnikov four-body problem when all primaries are oblate

- Physics, Geology
- 2013

We study the motions of an infinitesimal mass in the Sitnikov four-body problem in which three equal oblate spheroids (called primaries) symmetrical in all respect, are placed at the vertices of an…

### Nonplanar periodic solutions for spatial restricted 3-body and 4-body problems

- Mathematics
- 2012

In this paper, by using variational methods, we study the existence of nonplanar periodic solutions for the following spatial restricted 3-body and 4-body problems: for N=2 or 3$N=2 \mbox{ or } 3$, N…

### PYRAMIDAL CENTRAL CONFIGURATIONS AND PERVERSE SOLUTIONS

- Mathematics
- 2004

For n-body problems, a central configuration (CC) plays an im- portant role. In this paper, we establish the relation between the spatial pyramidal central configuration (PCC) and the planar central…

### Four-Body Central Configurations¶with some Equal Masses

- Mathematics
- 2002

Abstract We prove firstly that any convex non-collinear central configuration of the planar 4-body problem with equal opposite masses β >α > 0, such that the diagonal corresponding to the mass α is…

### Periodic orbits and bifurcations in the Sitnikov four-body problem

- Mathematics, Physics
- 2008

We study the existence, linear stability and bifurcations of what we call the Sitnikov family of straight line periodic orbits in the case of the restricted four-body problem, where the three equal…