Group Analysis of Differential Equations and Integrable Systems − 2018 Petre Birtea (West University of Timisoara, Romania) First and second order optimality conditions on the symplectic group Abstract: By embedding the symplectic group $Sp(2n,\mathbb{R})$ in $\mathbb{R}^{4n^2}$, we construct the gradient embedded vector field of a cost functional defined on the symplectic group, and we write it in the ambient coordinates. We present first and second order conditions for critical points of the cost functional. We further give an explicit formula for the Hessian operator on the symplectic group written in the ambient coordinates.