In The Variational Principles of Mechanics by Lanczos, in section 1 of Chapter 1, Lanczos states that for a complicated situation, the Newtonian approach fails to give a unique answer to the problem, in contrast to the analytical mechanics approach.
Can anyone provide an example where the Newtonian approach cannot give a unique solution to a mechanics problem? I don't mean a trivial response like simply re-expressing a solution in a different inertial frame, but a "complicated" situation that Lanczos had in mind. I cannot think of any.
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