  Following Jacobi's geometrization of Lagrange's least action principle,
trajectories of classical mechanics can be characterized as geodesics on the
configuration space M with respect to a suitable metric which is the conformal
modification of the kinematic metric by a factor (U+h), where U and h is the
potential function and total energy, respectively.In the special case of 3-body
motions with zero angular momentum, the global geometry of such trajectories
can be reduced to that of their moduli curves, which record the change of size
and shape, in the moduli space of oriented m-triangles, whose kinematic metric
is, in fact, a Riemannian cone over the shape space M*(a 2-sphere of radius
1/2). In this paper, we show that the moduli curve of such a motion is uniquely
determined by its shape curve (which only records the change of shape)when h is
nonzero, while in the special case of h=0 it is uniquely determined up to
scaling. Thus, the study of the global geometry of such motions can be further
reduced to that of the shape curves, which are time-parametrized curves on the
2-sphere characterized by a third order ODE (cf. Theorem 3.9). Moreover, these
curves have two remarkable properties, namely the uniqueness of parametrization
and the monotonicity (cf. Theorem 4.6 and Theorem 5.8) that constitute a solid
foundation for a systematic study of their global geometry and naturally lead
to the formulation of some pertinent problems such as those briefly discussed
in Section 7.
