Given a symplectic manifold, does it make sense to talk about "symplectic Hessian" at a critical point?

Question

If $f$ is a smooth function on a symplectic manifold $(M, \omega)$ we can define its symplectic gradient : this is a vector field $X_f$ such that $\iota_{X_f} \omega = - df$. My question is the following : consider a critical point of $f$, does it makes sense to talk about "symplectic Hessian" at this point ? If yes, how can we define/compute it ?