Abstract
This paper discusses several numerical approaches for solving problems arising in optimizing trajectories. The basic concepts underlying the gradient method, the second variation method, and a generalized Newton-Raphson method are presented in a very elementary manner by considering an ordinary minimum problem with a side constraint. The results obtained when the basic concepts are extended to the variational problem and the computational algorithms are then discussed. Finally, in the concluding remarks, advantages and disadvantages of each method are reviewed, and a comparison is made between the second variation method, which might be considered a direct method, and the generalized Newton-Raphson method, normally considered as an indirect method. Part II of this paper provides an application of the three methods to a specific problem.
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