3 ms·
I read them. The first paper you referenced do not address any points (it just gives another example, similar to the Norton's Dome), and the second one actually
by Miiko 5y ago
I read them. The first paper you referenced do not address any points (it just gives another example, similar to the Norton's Dome), and the second one actually confirms the major point given by Gruff Davies:
> Newton’s laws are deterministic, but they’re not complete.
That second paper identifies Lipschitz condition as missing part, and, by the way, it also states in the abstract:
> I do not seek to conclude that these examples are necessarily strong evidence that classical mechanics is not
deterministic; rather, I want to emphasize the legitimacy of pragmatic considerations in deciding what legitimately counts as a Newtonian system
That is, the original Newton's principle of determinacy ("The initial positions and velocities of all the particles of a mechanical system uniquely determine all of its motion.") might be proven wrong, but that does not make "classical" (non-relativistic) mechanics indeterministic, only incomplete. To quote Gruff Davies again:
> If we think about particles’ states, and consider higher orders like jounce, snap, crackle and pop. (and all the way to infinity), we can see that the choice of path of unstable particles is fully determined by their values, so this isn’t evidence for indeterminism, it is evidence for incompletion.
- deltasixeight 5y agoIncompleteness implies indeterminacy. Davies point is sort of pedantic but the math from Nortons paper is nondeterministic BECAUSE of incompleteness. We know at the singularity newtons laws are incomplete so in that region you are correct. Prior to the particle entering a singularity newtons laws describe it deterministically so you are still correct. At some unknown time when the particle exits the singularity Newtons laws still apply but are no longer deterministic, because we do not know what happened in the singularity. We do know the possible states of the particle are still bounded and controlled by newtons laws but within this boundary we are unable to fully determine its unique path if one should exist.
- Miiko 5y agoWell, if you formulate the statement as: > We do know that Newtonian mechanics is not deterministic at singularity points then I fully agree with that. However, that may be fixed by either adding additional requirement (e.g. Lipschitz continuity) or just by not considering Newtonian mechanics applicable to those cases - it is known already that Newton's laws do not fully describe the real word (because quantum uncertainty does exist) and the Lebesgue measure of singularity cases is zero anyway. In case of three-body problem, the singularities are the case of bodies collisions and yes, those cases are not deterministic, but the configuration without collision is known to be fully deterministic.