Optimality with respect to blips

We consider local minimality with respect to short duration variations, called ‘blips’. It is shown that for quadratic differential integrals either there are no such optimal trajectories, or that all stationary trajectories are local minima with respect to blips. Conditions on the polynomial matrix that defines the quadratic integral for the stationary trajectories to be local minima with respect to blips are derived. We motivate this problem by the variational principles of mechanics, and show that if the Hessian of the Lagrangian with respect to the generalized velocities is positive definite, then the solutions of the Euler-Lagrange equations are the local minimum of the action integral w.r.t. blips as variations.