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Article overview
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No hyperbolic pants for the 4-body problem | Connor Jackman
; Richard Montgomery
; | Date: |
2 Feb 2015 | Abstract: | The $N$-body problem with a $1/r^2$ potential has, in addition to translation
and rotational symmetry, an effective scale symmetry which allows its zero
energy flow to be reduced to a geodesic flow on complex projective $N-2$-space,
minus a hyperplane arrangement. When $N=3$ we get a geodesic flow on the
two-sphere minus three points. If, in addition we assume that the three masses
are equal, then it was proved in [1] that the corresponding metric is
hyperbolic: its Gaussian curvature is negative except at two points. Does the
negative curvature property persist for $N=4$, that is, in the equal mass
$1/r^2$ 4-body problem? Here we prove ’no’ by computing that the corresponding
Riemannian metric in this $N=4$ case has positive sectional curvature at some
two-planes. This ’no’ answer dashes hopes of naively extending hyperbolicity
from $N=3$ to $N>3$. | Source: | arXiv, 1502.0606 | Services: | Forum | Review | PDF | Favorites |
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