Symmetric Periodic Solutions for the Spatial Maxwell Restricted $N+1$-Problem with Manev Potential

Abstract

In the spatial Maxwell restricted $N+1$-body problem, the motion of an infinitesimal particle attracted by the gravitational field of (N) bodies is studied. These bodies are arranged in a planar ring configuration. This configuration consists of $N-1$ primaries of equal mass m located at the vertices of a regular polygon that is rotating on its own plane about its center of mass with a constant angular velocity $\omega$. Another primary of mass $m_0=\beta m$ ($\beta >0$ parameter) is placed at the center of the ring. Moreover, we assume that the central body may be an ellipsoid, or radiation source, which introduces a new parameter e. The existence of several families of symmetric periodic solutions for the spatial Maxwell restricted $(N+1)$-problem with Manev potential is proved. More precisely, firstly we get symmetric periodic solutions around the central body (attractor or repulsor) close to the equatorial plane and small parameter of oblateness. Secondly, we obtain symmetric periodic solutions far away of the central body and peripherals, close to the equatorial plane with arbitrary oblateness. Furthermore, all these families of periodic solutions are stable.

Appendix

We will present the expressions of the partial derivatives associated with the disturbed function $H_1$ in (3.6), in terms of Poincaré-Delaunay variables (3.5). We use the Mathematica processor to get them.

$$\begin{aligned} \frac{\partial {\mathcal {H}}_1}{\partial P_1}= & {} -\delta _1+\delta _2 \frac{C}{4P_1^7}\left( -16 P_1^2+\frac{4 P_1 Q_2 \sin (Q_1) \left( 5 \left( P_2^2+Q_2^2\right) -18 P_1\right) }{\sqrt{4 P_1-P_2^2-Q_2^2}}\right. \\&+5 (P_2-Q_2) (P_2+Q_2) \cos (2 Q_1) \left( 3 \left( P_2^2+Q_2^2\right) -10 P_1\right) \\&+4 P_2 \cos (Q_1) \left( 5 Q_2 \sin (Q_1) \left( 3 \left( P_2^2+Q_2^2\right) -10 P_1\right) +\frac{P_1 \left( 5 \left( P_2^2+Q_2^2\right) -18 P_1\right) }{\sqrt{4 P_1-P_2^2-Q_2^2}}\right) \\&-10 P_1 \left( P_2^2+Q_2^2\right) +3 \left( P_2^2+Q_2^2\right) ^2),\\ \frac{\partial {\mathcal {H}}_1}{\partial P_2}= & {} \delta _1 P_2+\delta _2\frac{C}{8P_1^6}(-10 Q_2 \sin (2 Q_1) \left( -4 P_1+3 P_2^2+Q_2^2\right) \\&-\frac{8 P_1 P_2 Q_2 \sin (Q_1)}{\sqrt{4 P_1-P_2^2-Q_2^2}}\\&+\frac{8 P_1 \cos (Q_1) \left( 4 P_1-2 P_2^2-Q_2^2\right) }{\sqrt{4 P_1-P_2^2-Q_2^2}}-20 P_2 \left( P_2^2-2 P_1\right) \cos (2 Q_1)\\&-4 P_2 \left( -2 P_1+P_2^2+Q_2^2\right) ),\\ \frac{\partial {\mathcal {H}}_1}{\partial P_3}= & {} \delta _1 P_3,\\ \frac{\partial {\mathcal {H}}_1}{\partial Q_1}= & {} \delta _2 \frac{C}{2P_1^6}(Q_2 \cos (Q_1)-P_2 \sin (Q_1)) \left( 2 P_1 \sqrt{4 P_1-P_2^2-Q_2^2}\right. \\&\left. -5 \left( -4 P_1+P_2^2+Q_2^2\right) (P_2 \cos (Q_1)+Q_2 \sin (Q_1))\right) ,\\ \frac{\partial {\mathcal {H}}_1}{\partial Q_2}= & {} \delta _1 Q_2+\delta _2\frac{C}{8P_1^6}\left( -10 P_2 \sin (2 Q_1) \left( -4 P_1+P_2^2+3 Q_2^2\right) \right. \\&-\frac{8 P_1 \sin (Q_1) \left( -4 P_1+P_2^2+2 Q_2^2\right) }{\sqrt{4 P_1-P_2^2-Q_2^2}}\\&\left. -\frac{8 P_1 P_2 Q_2 \cos (Q_1)}{\sqrt{4 P_1-P_2^2-Q_2^2}}-4 Q_2 \left( -2 P_1+P_2^2+Q_2^2\right) +20 Q_2 \left( Q_2^2-2 P_1\right) \cos (2 Q_1)\right) \\ \frac{\partial {\mathcal {H}}_1}{\partial Q_3}= & {} \delta _1 Q_3. \end{aligned}$$