New Classes of Spatial Central Configurations forN+N+ 2-Body Problem

Abstract

Under arbitrary masses, in this paper, we discuss the existence of new families of spatial central configurations for the <i >N</i> + <i >N</i> + 2-body problem, <svg style="vertical-align:-1.29163pt;width:40.625px;" id="M1" height="12.3" version="1.1" viewBox="0 0 40.625 12.3" width="40.625" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(1.25,0,0,-1.25,0,12.3)"> <g transform="translate(72,-62.16)"> <text transform="matrix(1,0,0,-1,-71.95,63.5)"> <tspan style="font-size: 12.50px; " x="0" y="0">𝑁</tspan> <tspan style="font-size: 12.50px; " x="14.103384" y="0">≥</tspan> <tspan style="font-size: 12.50px; " x="26.143772" y="0">2</tspan> </text> </g> </g> </svg>. We study some necessary conditions and sufficient conditions for a families of spatial double pyramidical central configurations (d.p.c.c.), where 2<i >N</i> bodies are at the vertices of a nested regular <i >N</i>-gons <svg style="vertical-align:-3.13504pt;width:47.299999px;" id="M2" height="14.375" version="1.1" viewBox="0 0 47.299999 14.375" width="47.299999" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(1.25,0,0,-1.25,0,14.375)"> <g transform="translate(72,-60.5)"> <text transform="matrix(1,0,0,-1,-71.95,63.67)"> <tspan style="font-size: 12.50px; " x="0" y="0">Γ</tspan> </text> <text transform="matrix(1,0,0,-1,-64.61,60.55)"> <tspan style="font-size: 8.75px; " x="0" y="0">1</tspan> </text> <text transform="matrix(1,0,0,-1,-56.96,63.67)"> <tspan style="font-size: 12.50px; " x="0" y="0">∪</tspan> <tspan style="font-size: 12.50px; " x="10.527526" y="0">Γ</tspan> </text> <text transform="matrix(1,0,0,-1,-39.09,60.55)"> <tspan style="font-size: 8.75px; " x="0" y="0">2</tspan> </text> </g> </g> </svg>, and the other two bodies are symmetrically located on the straight line that is perpendicular to the plane that contains <svg style="vertical-align:-3.13504pt;width:47.299999px;" id="M3" height="14.375" version="1.1" viewBox="0 0 47.299999 14.375" width="47.299999" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(1.25,0,0,-1.25,0,14.375)"> <g transform="translate(72,-60.5)"> <text transform="matrix(1,0,0,-1,-71.95,63.67)"> <tspan style="font-size: 12.50px; " x="0" y="0">Γ</tspan> </text> <text transform="matrix(1,0,0,-1,-64.61,60.55)"> <tspan style="font-size: 8.75px; " x="0" y="0">1</tspan> </text> <text transform="matrix(1,0,0,-1,-56.96,63.67)"> <tspan style="font-size: 12.50px; " x="0" y="0">∪</tspan> <tspan style="font-size: 12.50px; " x="10.527526" y="0">Γ</tspan> </text> <text transform="matrix(1,0,0,-1,-39.09,60.55)"> <tspan style="font-size: 8.75px; " x="0" y="0">2</tspan> </text> </g> </g> </svg> and passes through the geometric center of <svg style="vertical-align:-3.13504pt;width:47.299999px;" id="M4" height="14.375" version="1.1" viewBox="0 0 47.299999 14.375" width="47.299999" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(1.25,0,0,-1.25,0,14.375)"> <g transform="translate(72,-60.5)"> <text transform="matrix(1,0,0,-1,-71.95,63.67)"> <tspan style="font-size: 12.50px; " x="0" y="0">Γ</tspan> </text> <text transform="matrix(1,0,0,-1,-64.61,60.55)"> <tspan style="font-size: 8.75px; " x="0" y="0">1</tspan> </text> <text transform="matrix(1,0,0,-1,-56.96,63.67)"> <tspan style="font-size: 12.50px; " x="0" y="0">∪</tspan> <tspan style="font-size: 12.50px; " x="10.527526" y="0">Γ</tspan> </text> <text transform="matrix(1,0,0,-1,-39.09,60.55)"> <tspan style="font-size: 8.75px; " x="0" y="0">2</tspan> </text> </g> </g> </svg>. We prove that if the bodies are in a d.p.c.c., then the masses on each <i >N</i>-gon are equal, and the other two are also equal. And also we prove the existence and uniqueness of the central configurations for any given ratios of masses.