  An amplitude analysis of the $D^+ \to \pi^- \pi^+ \pi^+$ decay is performed
with a sample corresponding to $1.5\rm fb^{-1}$ of integrated luminosity of
$pp$ collisions at a centre-of-mass energy $\sqrt{s}=8$ TeV collected by the
LHCb detector in 2012. The sample contains approximately six hundred thousand
candidates with a signal purity of $95\%$. The resonant structure is studied
through a fit to the Dalitz plot where the $\pi^- \pi^+$ S-wave amplitude is
extracted as a function of $\pi^-\pi^+$ mass, and spin-1 and spin-2 resonances
are included coherently through an isobar model. The S-wave component is found
to be dominant, followed by the $\rho(770)^0\pi^+$ and $f_2(1270)\pi^+$
components. A small contribution from the $\omega(782)\to\pi^-\pi^+$ decay is
seen for the first time in the $D^+ \to \pi^- \pi^+ \pi^+$ decay.
