On the Problem of Infinite Spin in Total Collisions of the Planar 𝑁-Body Problem

For the planar <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper N"> <mml:semantics> <mml:mi>N</mml:mi> <mml:annotation encoding="application/x-tex">N</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-body problem, we introduce a class of moving coordinates suitable for orbits near central configurations, especially for total collision orbits, which is the main new ingredient of this paper. The moving coordinates allow us to reduce the degeneracy of the <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper N"> <mml:semantics> <mml:mi>N</mml:mi> <mml:annotation encoding="application/x-tex">N</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-Body problem from its intrinsic symmetrical characteristic. First, we give a full answer to the infinite spin or <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper P a i n l e v ModifyingAbove e With acute"> <mml:semantics> <mml:mrow> <mml:mi>P</mml:mi> <mml:mi>a</mml:mi> <mml:mi>i</mml:mi> <mml:mi>n</mml:mi> <mml:mi>l</mml:mi> <mml:mi>e</mml:mi> <mml:mi>v</mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mover> <mml:mi>e</mml:mi> <mml:mo>´</mml:mo> </mml:mover> </mml:mrow> </mml:mrow> <mml:annotation encoding="application/x-tex">Painlev\acute {e}</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-<inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper W i n t n e r"> <mml:semantics> <mml:mrow> <mml:mi>W</mml:mi> <mml:mi>i</mml:mi> <mml:mi>n</mml:mi> <mml:mi>t</mml:mi> <mml:mi>n</mml:mi> <mml:mi>e</mml:mi> <mml:mi>r</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">Wintner</mml:annotation> </mml:semantics> </mml:math> </inline-formula> problem in the case corresponding to nondegenerate central configurations. Then following some original ideas of C.L. Siegel, especially the idea of normal forms, and applying the theory of central manifolds, we give a partial answer to the problem in the case corresponding to degenerate central configurations. We completely answer the problem in the case corresponding to central configurations with degree of degeneracy one. Combining some results on the planar nonhyperbolic equilibrium point, we give a criterion in the case corresponding to central configurations with degree of degeneracy two. We further answer the problem in the case corresponding to all known central configurations of four bodies. Therefore, we solve the problem for almost every choice of the masses of the four-body problem. Finally, we give a measure of the set of initial conditions leading to total collisions.