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Recently i've learned the formulation of Gauss's principle of least constraints, which states that the motion of a system of material points is in maximal accordance with free motion, or under least constraint, where the constraint is measured as: $$Z = \Sigma_{k = 1}^{N} m_k(\frac {{d^2r_k}}{{dt^2}} - \frac {{F_k}}{{m_k}})^2.$$

I'm trying to find an example of a problem in classical mechanics which cannot be solved at ease by the Newtonian formalism but can be solved easily (relative to the Newtonian approach) using this principle. Such an example can help me understand the motivation behind the formulation of the principle.

If anybody helps, please give a specific example (with solution) and not just general remarks.

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