  We consider the $N$-body problem in $\mathbb{R}^d$ with the newtonian
potential $1/r$. We prove that for every initial configuration $x_i$ and for
every minimizing normalized central configuration $x_0$, there exists a
collision-free parabolic solution starting from $x_i$ and asymptotic to $x_0$.
This solution is a minimizer in every time interval. The proof exploits the
variational structure of the problem, and it consists in finding a convergent
subsequence in a family of minimizing trajectories. The hardest part is to show
that this solution is parabolic and asymptotic to $x_0$.
