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

For the planar $N$-body problem, we first introduce a class of moving frame suitable for orbits near central configurations, especially for total collision orbits, which is the main new ingredient of this paper. The moving frame allows us to reduce the degeneracy of the problem according to intrinsic symmetrical characteristic of the $N$-Body problem. First, we give a full answer to the infinite spin or $Painlev\acute{e}$-$Wintner$ 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 one degree of degeneracy. Combining some results on the planar nonhyperbolic equilibrium point, we give a criterion for the case corresponding to central configurations with two degrees of degeneracy. We further answer the problem in the case corresponding to all known degenerate central configurations of four-body. 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.