HOLDITCH-TYPE THEOREMS FOR THE POLAR MOMENT OF INERTIA UNDER THE 1-PARAMETER CLOSED PLANAR HOMOTHETIC MOTION

In this study, during the 1-parameter closed homothetic motion, the Holditch-Type Theorems are presented for the polar moments of inertia of the closed orbit curves of three non-collinear points.

On the free time minimizers of the NewtonianN-body problem

In this paper we study the existence and the dynamics of a very special class of motions, which satisfy a strong global minimization property. More precisely, we call a free time minimizer a curve which satisfies the least action principle between any pair of its points without the constraint of time for the variations. An example of a free time minimizer defined on an unbounded interval is a parabolic homothetic motion by a minimal central configuration. The existence of a large amount of free time minimizers can be deduced from the weak KAM theorem. In particular, for any choice ofx0, there should be at least one free time minimizerx(t)defined for allt≥ 0 and satisfyingx(0)=x0. We prove that such motions are completely parabolic. Using Marchal's theorem we deduce as a corollary that there are no entire free time minimizers, i.e. defined on$\mathbb{R}$. This means that the Mañé set of the NewtonianN-body problem is empty.