Theories and Inequalities on the Satellite System

We define the satellite system without any central, the satellite system with a central, and the satellite system of single point with a central. For the satellite system without any central S { Γ , Γ N } , we establish the inequality:∮ Γ d s ∮Γ N| P Q | d s Q ≤ ( N | Γ | 3 / 4 π 2)sin ⁡ ( 2 π / N ) . For the satellite system with a central S { O , Γ , Γ N } , we establish the following inequality under the proper hypothesis:∮ ΓM N [ t ] ( r ) d s ≤ (| Γ |2 / 2 π )cos ⁡( π / N ) ( ∀ t ∈ ] - ∞ , - 2 ] ) . As an application, we get the inequalities∮ Γρ P d s ≥ ∮ Γr P d s ≥ 2 | D ( Γ ) | , for the satellite system of single point with a central S { O , Γ , P } . For these results, there are generalized backgrounds in the fields of differential geometry and space science.