Non-uniqueness of solutions in Newtonian mechanics

Question

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.

Answer 1

I don't know what Lanczos had in mind, but Norton's dome is an example of nonuniqueness. There is a large literature on this system and its philosophical implications (or lack thereof).

• Norton, "Causation as Folk Science," http://philsci-archive.pitt.edu/1214/

• Korolev, "Indeterminism, asymptotic reasoning, and time irreversibility in classical physics," 2006, http://philsci-archive.pitt.edu/3003/

• Jon Pérez Laraudogoitia, "On Norton’s dome," Synthese, 2012, http://link.springer.com/content/pdf/10.1007%2Fs11229-012-0105-z

• Fletcher, What Counts as a Newtonian System? The View from Norton’s Dome

• David Malament, Norton's Slippery Slope, Philosophy of Science 75 (5):799-816 (2008)

Answer 2

I'd say fluid mechanics problems. You might want to check Navier-Stokes equations which are basically derived from a newtonian perspective.