Equivalence of rigid motions and relative equilibria in the <i>N</i>-body problem on the two-sphere

Abstract We investigate the relationship between rigid motions and relative equilibria in the N -body problem on the two-dimensional sphere, <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mrow> <mml:msup> <mml:mrow> <mml:mi mathvariant="double-struck">S</mml:mi> </mml:mrow> <mml:mn>2</mml:mn> </mml:msup> </mml:mrow> </mml:math> . We prove that any rigid motion of the N -body system on <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mrow> <mml:msup> <mml:mrow> <mml:mi mathvariant="double-struck">S</mml:mi> </mml:mrow> <mml:mn>2</mml:mn> </mml:msup> </mml:mrow> </mml:math> must be a relative equilibrium. Our approach extends the classical study of rigid body dynamics and utilizes a rotating frame attached to the particles to derive the corresponding equations of motion. We further show that our results can be extended to the N -body gravitational system in <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mrow> <mml:msup> <mml:mrow> <mml:mi mathvariant="double-struck">R</mml:mi> </mml:mrow> <mml:mn>3</mml:mn> </mml:msup> </mml:mrow> </mml:math> . The results are oriented to a broader understanding of the dynamics of N -body systems on curved surfaces.