Feedback-Invariant Optimal Control Theory and Differential Geometry -I. Regular Extremals A. A. Agrachev, R. V. Gamkrelidze Abstract: Symplectic approach to smooth optimal control problems and some geometric structures is considered. A Hamiltonian method of investigating regular extremals is developed, leading to a far-reaching generalization of the differential-geometric method of investigation Riemannian geodesics in terms of the Levi-Civita connection and the curvature tensor.