The numerical implementation process can be distilled into a gradient search within a finite-dimensional approximation of the path space. One approach involves discretizing the path at uniform time intervals. Another effective method entails expressing the action as a function of the Fourier components of the choreography curve and truncating the Fourier series at a finite order. Convergence can be verified by increasing the truncation order and observing the alterations in the resulting minimizing Fourier coefficients. If these changes fall within a specified tolerance range, the result can be considered successful. For further refinements, consult the works of Simó or Moore and Nauenberg.