Second-order necessary conditions for the infinite-horizon variational problems

We give new proofs of first-order and second-order necessary conditions for the infinite-horizon variational problems in the framework of the continuous time. We study the notion of conjugate points and we build new second-order necessary conditions. We also translate all of our results into the Hamiltonian formalism. Moreover, we apply these general results to the special Lagrangians in the form e^−δt L(x, x).