  We regard the real symplectic group $Sp(2n,\mathbb{R})$ as a constraint
submanifold of the $2n\times 2n$ real matrices $\mathcal{M}_{2n}(\mathbb{R})$
endowed with the Euclidean (Frobenius) metric, respectively as a submanifold of
the general linear group $Gl(2n,\mathbb{R})$ endowed with the (left) invariant
metric. For a cost function that defines an optimization problem on the real
symplectic group we give a necessary and sufficient condition for critical
points and we apply this condition to the particular case of a least square
cost function. In order to characterize the critical points we give a formula
for the Hessian of a cost function defined on the real symplectic group, with
respect to both considered metrics. For a generalized Brockett cost function we
present a necessary condition and a sufficient condition for local minimum. We
construct a retraction map that allows us to detail the steepest descent and
embedded Newton algorithms for solving an optimization problem on the real
symplectic group.
