  We report the first measurement of the full angular distribution for
inclusive $J/\psi\rightarrow\mu^{+}\mu^{-}$ decays in $p$$+$$p$ collisions at
$\sqrt{s}=510$ GeV. The measurements are made for $J/\psi$ transverse momentum
$2<p_{T}<10$ GeV/$c$ and rapidity $1.2<y<2.2$ in the Helicity, Collins-Soper,
and Gottfried-Jackson reference frames. In all frames the polar coefficient
$\lambda_{\theta}$ is strongly negative at low $p_{T}$ and becomes close to
zero at high $p_{T}$, while the azimuthal coefficient $\lambda_{\phi}$ is close
to zero at low $p_{T}$, and becomes slightly negative at higher $p_{T}$. The
frame-independent coefficient $\tilde{\lambda}$ is strongly negative at all
$p_{T}$ in all frames. The data are compared to the theoretical predictions
provided by nonrelativistic quantum chromodynamics models.
