We develop a Newton method for the optimization of trajectory functionals. Through the use of a trajectory tracking nonlinear projection operator, the dynamically constrained optimization problem is converted into an unconstrained problem, making many aspects of the algorithm rather transparent. Examples: first and second order optimality conditions, search direction and step length calculations, update rule—all developed from an unconstrained point of view. Quasi-Newton methods are easily developed as well, allowing straightforward globalization of the Newton method. As all operations are set in an appropriate Banach space, properties such as solution regularity are retained so that implementation decisions (level of discretation, etc.) are based on approximating the solution rather than the problem. Convergence in Banach space is shown to be quadratic as is usual for Newton methods. Keywords: nonlinear optimal control, trajectory optimization, Newton methods, Banach manifolds, nonlinear projection operators, trajectory manifold.