The Kepler problem is the special case $\alpha = 1$ of the power law problem: to solve Newton's equations for a central force whose potential is of the form $-\mu/r^{\alpha}$ where $\mu$ is a coupling constant. Associated to such a problem is a two-dimensional cone with cone angle $2 \pi c$ with $c = 1 - \frac{\alpha}{2}$. We construct a transformation taking the geodesics of this cone to the zero energy solutions of the $\alpha$-power law problem. The `Kepler Cone' is the cone associated to the Kepler problem. This zero-energy cone transformation is a special case of a transformation discovered by Maclaurin in the 1740s transforming the $\alpha$- power law problem for any energies to a `Maclaurin dual' $\gamma$-power law problem where $\gamma = \frac{2 \alpha}{2-\alpha}$ and which, in the process, mixes up the energy of one problem with the coupling constant of the other. We derive Maclaurin duality using the Jacobi-Maupertuis metric reformulation of mechanics. We then use the conical metric to explain properties of Rutherford-type scattering off power law potentials at positive energies. The one possibly new result in the paper concerns ``star-burst curves'' which arise as limits of families negative energy solutions as their angular momentum tends to zero. We relate geodesic scattering on the cone to Rutherford type scattering of beams of solutions in the potential. We describe some history around Maclaurin duality and give two derivations of the Jacobi-Maupertuis metric reformulation of classical mechanics. The piece is expository, aimed at an upper-division undergraduate. Think American Math. Monthly.

Attached to a singular analytic curve germ in $d$-space is a numerical semigroup: a subset $S$ of the non-negative integers which is closed under addition and whose complement isfinite. Conversely, associated to any numerical semigroup $S$ is a canonical mononial curve in $e$-space where $e$ is the number of minimal generators of the semigroup. It may happen that $d < e = e(S)$ where $S$ is the semigroup of the curve in $d$-space. Define the minimal (or `honest') embedding of a numerical semigroup to be the smallest $d$ such that $S$ is realized by a curve in $d$-space. Problem: characterize the numerical semigroups having minimal embedding dimension $d$. The answer is known for the case $d=2$ of planar curves and reviewed in an Appendix to this paper. The case $d =3$ of the problem is open. Our main result is a characterization of the multiplicity $4$ numerical semigroups whose minimal embedding dimension is $3$. See figure 1. The motivation for this work came from thinking about Legendrian curve singularities.

The subject is brake orbits for the 3-body problem: orbits where all velocities are zero at some instant. We extract a paradox and a mystery out of a recent database of 30 non-collision periodic brake orbits for the equal mass 3-body problem built up by Li and Liao. We solve the paradox and the mystery and pose an open question.

For the Newtonian 4-body problem in space we prove that any zero angular momentum bounded solution suffers infinitely many coplanar instants, that is, times at which all 4 bodies lie in the same plane. This result generalizes a known result for collinear instants ("syzygies") in the zero angular momentum planar 3-body problem, and extends to the $d+1$ body problem in $d$-space. The proof, for $d=3$, starts by identifying the center-of-mass zero configuration space with real $3 \times 3$ matrices, the coplanar configurations with matrices whose determinant is zero, and the mass metric with the Frobenius (standard Euclidean) norm. Let $S$ denote the signed distance from a matrix to the hypersurface of matrices with determinant zero. The proof hinges on establishing a harmonic oscillator type ODE for $S$ along solutions. Bounds on inter-body distances then yield an explicit lower bound $\omega$ for the frequency of this oscillator, guaranteeing a degeneration within every time interval of length $\pi/\omega$. The non-negativity of the curvature of oriented shape space (the quotient of configuration space by the rotation group) plays a crucial role in the proof.

The Jacobi-Maupertuis metric provides a reformulation of the classical N-body problem as a geodesic flow on an energy-dependent metric space denoted $M_E$ where $E$ is the energy of the problem. We show that $M_E$ has finite diameter for $E < 0$. Consequently $M_E$ has no metric rays. Motivation comes from work of Burgos- Maderna and Polimeni-Terracini for the case $E \ge 0$ and from a need to correct an error made in a previous ``proof''. We show that $M_E$ has finite diameter for $E < 0$ by showing that there is a constant $D$ such that all points of the Hill region lie a distance $D$ from the Hill boundary. (When $E \ge 0$ the Hill boundary is empty.) The proof relies on a game of escape which allows us to quantify the escape rate from a closed subset of configuration space, and the reduction of this game to one of escaping the boundary of a polyhedral convex cone into its interior.

For n bodies moving in Euclidean d-space under the influence of a homogeneous pair interaction we compactify every center-of-mass energy surface, obtaining a 2d(n -1)-1 - dimensional manifold with corners in the sense of Melrose. After a time change, the flow on this manifold is globally defined and non-trivial on the boundary.