Expansive solutions and the boundary at infinity for the homogeneous $N$-body problem
We investigate expansive solutions of the $N$-body problem in $\mathbb{R}^d$ ($d\ge2$) driven by homogeneous Newtonian potentials of degree $-\alpha$. We establish the existence of half-entire expansive motions with prescribed initial configuration and asymptotic direction for a wide range of homogeneity exponents $\alpha$. Our approach is variational and relies on the minimization of a suitably renormalized Lagrangian action, allowing us to treat in a unified framework the hyperbolic, parabolic, and hyperbolic-parabolic regimes in the sense of Chazy's classification. Beyond existence, we derive refined asymptotic expansions for all classes of expansive solutions, identifying higher-order correction terms and improving previously known growth estimates, including the classical Newtonian case $\alpha=1$. In particular, for hyperbolic-parabolic solutions, we provide a detailed description of the interplay between linear escape of cluster centers and internal parabolic dynamics, extending the cluster scattering picture to general homogeneous potentials. Finally, we interpret these solutions within the geometric framework of the Jacobi-Maupertuis metric and the weak KAM theory. In this perspective, expansive motions correspond to geodesic rays and calibrating curves for the associated Hamilton-Jacobi equation, yielding a dynamical characterization of the boundary at infinity and a refined description of global viscosity solutions.