Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Morse theory for $S$-balanced configurations in the Newtonian $n$-body
  problem   For the Newtonian (gravitational) $n$-body problem in the Euclidean
$d$-dimensional space, the simplest possible solutions are provided by those
rigid motions (homographic solutions) in which each body moves along a
Keplerian orbit and the configuration of the $n$-body is a constant up to
rotations and scalings named \textit{central configuration}. For $d\leq 3$, the
only possible homographic motions are those given by central configurations.
For $d \geq 4$ instead, new possibilities arise due to the higher complexity of
the orthogonal group $O(d)$, as observed by Albouy and Chenciner. For instance,
in $\mathbb R^4$ it is possible to rotate in two mutually orthogonal planes
with different angular velocities. This produces a new balance between
gravitational forces and centrifugal forces providing new periodic and
quasi-periodic motions. So, for $d\geq 4$ there is a wider class of
$S$-\textit{balanced configurations} (containing the central ones) providing
simple solutions of the $n$-body problem, which can be characterized as well
through critical point theory. In this paper, we first provide a lower bound on
the number of balanced (non-central) configurations in $\mathbb R^d$, for
arbitrary $d\geq 4$, and establish a version of the $45^\circ$-theorem for
balanced configurations, thus answering some questions raised by Moeckel. Also,
a careful study of the asymptotics of the coefficients of the Poincar\'e
polynomial of the collision free configuration sphere will enable us to derive
some rather unexpected qualitative consequences on the count of $S$-balanced
configurations. In the last part of the paper, we focus on the case $d=4$ and
provide a lower bound on the number of periodic and quasi-periodic motions of
the gravitational $n$-body problem which improves a previous celebrated result
of McCord.

5
The categories of this paper are: gr-qc
3
The categories of this paper are: physics
25
The categories of this paper are: patt-sol
20
The categories of this paper are: acc-phys
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The categories of this paper are: dg-ga
6
The categories of this paper are: astro-ph
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The categories of this paper are: math-ph
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The categories of this paper are: q-bio
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The categories of this paper are: nlin
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The categories of this paper are: solv-int
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The categories of this paper are: stat
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The categories of this paper are: cmp-lg
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The categories of this paper are: q-fin
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The categories of this paper are: mtrl-th
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The categories of this paper are: q-alg
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The categories of this paper are: plasm-ph