In this paper, the necessary and sufficient conditions for minima plane path with a movable end-point are developed. Using the calculus of variations the considered conditions are based on the zero first-order nonsimultaneous variation and on the positive second-order variation in the functional of integral type corresponding to mechanical systems. The applied procedure is the coordinate parametric method. The obtained solutions are tested on a brachistochrone with one end-point constrained to lie on a circle. The exact solution is compared with the approximate one obtained with Ritz’s method.
The Brachistochrone With a Movable End-Point and the Nonsimultaneous Variations
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Radomirovic, D., Djukic, D., and Cveticanin, L. (November 18, 2009). "The Brachistochrone With a Movable End-Point and the Nonsimultaneous Variations." ASME. J. Comput. Nonlinear Dynam. January 2010; 5(1): 011007. https://doi.org/10.1115/1.4000316
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