Fixed-energy initial-condition optimality ($P_H\nabla J({\mathbf{u}}_0)=\lambda {\mathbf{u}}_0$)

For a differentiable objective on an admissible real Hilbert space $H$, with $\Vert {\mathbf{u}}_0{\Vert }^2/2=E_0>0$, stationarity on the sphere in a normed vector space of fixed kinetic energy means that the projected gradient is normal to the sphere in a normed vector space. Thus $P_H\nabla J=\lambda {\mathbf{u}}_0$ for a real Lagrange multiplier. It follows by testing all tangent variations orthogonal to ${\mathbf{u}}_0$. The multiplier need not be positive; second-order conditions are still needed for a local minimum.