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For the planar $N$-body problem, we first introduce a class of moving framesuitable for orbits near central configurations, especially for total collisionorbits, which is the main new ingredient of this paper. The moving frame allowsus to reduce the degeneracy of the problem according to intrinsic symmetricalcharacteristic of the $N$-Body problem. First, we give a full answer to theinfinite spin or $Painlev\acute{e}$-$Wintner$ problem in the case correspondingto nondegenerate central configurations. Then following some original ideas ofC.L. Siegel, especially the idea of normal forms, and applying the theory ofcentral manifolds, we give a partial answer to the problem in the casecorresponding to degenerate central configurations. We completely answer theproblem in the case corresponding to central configurations with one degree ofdegeneracy. Combining some results on the planar nonhyperbolic equilibriumpoint, we give a criterion for the case corresponding to central configurationswith two degrees of degeneracy. We further answer the problem in the casecorresponding to all known degenerate central configurations of four-body.Therefore, we solve the problem for almost every choice of the masses of thefour-body problem. Finally, we give a measure of the set of initial conditionsleading to total collisions.

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