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623127 
Journal Article 
Global geometry of 3-body motions with vanishing angular momentum I 
Wu-Yi, H; Straume, E 
2008 
29 
1 
1-54 
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 the factor ( U + h), where U and h are the potential function and the 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 * ≃ S 2(1/2). In this paper, it is shown that the moduli curve of such a motion is uniquely determined by its shape curve (which only records the change of shape) in the case of h h ≠ 0, 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. Moreover, these curves have two remarkable properties, namely the uniqueness of parametrization and the monotonicity, that constitute a solid foundation for a systematic study of their global geometry and naturally lead to the formulation of some pertinent problems. [ABSTRACT FROM AUTHOR] Copyright of Chinese Annals of Mathematics is the property of Springer Science & Business Media B.V. and its content may not be copied or emailed to multiple sites or posted to a listserv without the copyright holder's express written permission. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract. (Copyright applies to all Abstracts) 
GEOMETRY, Differential; VARIATIONAL principles; WAVE functions; GEOMETRY; ANGULAR momentum (Mechanics); GEOMETRY, Plane; 3-Body problem; Kinematic geometry; Reduction; Shape curves