Global geometry of planary 3-body motions

作者：Hsiang Wu Yi; Straume Eldar^*

来源：Acta Applicandae Mathematicae, 2008, 101(1-3): 105-119.

DOI：10.1007/s10440-008-9198-2

• planary 3-body problem
• Riemannian geometric reduction
• separation of size and shape
• spherical shape curves

The planary 3-body problem is investigated in the framework of equivariant Riemannian geometry, where the global geometry of the trajectories of the 3-body motion are reduced to that of their moduli curves. These curves record the change of size and shape, in the 3-dimensional moduli space of oriented triangles with a given mass distribution. However, it is shown that the moduli curve, with some obvious exceptions, is already determined by the associated shape curve on the shape space M* similar or equal to S-2, which only records the change of the similarity class of the triangle. In this way the 3-body motion is encoded into the relative geometry between the shape curve gamma* and the gradient field del U* of the induced Newtonian potential function U* on the 2-sphere M*. In particular, a separation of size and shape is achieved, the size function can be reconstructed from gamma* and the latter is a solution of a 3rd order ODE on the 2-sphere.

• 出版日期2008-4