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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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    $\begingroup$ For non-determinism in Newtonian mechanics, see e.g. Norton's dome $\endgroup$
    – Qmechanic ♦
    Commented Oct 19, 2013 at 19:43
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    $\begingroup$ There are a variety of hard-contact problems in Newtonian mechanics which fail to give unique solutions. For example, the simultaneous collision of three spheres. Of course such problems are obviously a mathematical curiosity since they involve infinitely sharp contact potentials, but they are fun to think about. There are also some strange solutions that occur with certain initial conditions (such as a swinging pendulum given just enough energy to reach its apex), also of course unrealistic since the initial conditions have to be perfectly fine tuned. $\endgroup$
    – Nanite
    Commented Oct 19, 2013 at 21:00
  • $\begingroup$ Thanks, Nanite. I think this is along the lines of what Lanczos was getting at. Would you happen to know of any treatment of this problem (3 spheres scattering elastically)? I googled it and didn't find anything, and I've tried to work it myself and the math is getting pretty ugly. $\endgroup$
    – user31350
    Commented Oct 20, 2013 at 2:06
  • $\begingroup$ Another example is a ball colliding with two walls at once (i.e. an inside corner), where the angle $\theta$ between the walls is something other than $90^\circ$. Mathematically, two reflections in planes make a rotation by twice the angle between the planes, but the direction of rotation depends on the order of the two reflections. So the ball's velocity can turn by $2\theta$ either left or right. $\endgroup$
    – mr_e_man
    Commented Feb 2, 2023 at 18:39
  • $\begingroup$ But that's assuming that there are actually two collisions happening in succession. If it's considered as a single collision, then the direction of the force on the ball is not determined; it could be anywhere between the two walls' normal vectors. $\endgroup$
    – mr_e_man
    Commented Feb 2, 2023 at 18:47

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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).

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  • $\begingroup$ Thanks for pointing me to these references, Norton's dome is a new concept for me to think about. $\endgroup$
    – user31350
    Commented Oct 20, 2013 at 2:09
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I'd say fluid mechanics problems. You might want to check Navier-Stokes equations which are basically derived from a newtonian perspective.

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  • $\begingroup$ Can you explain what you mean? Under what circumstances are the solutions non-unique? $\endgroup$
    – user4552
    Commented Oct 19, 2013 at 21:07
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    $\begingroup$ I guess I should have been more clear on my answer. It is still an open question whether the solutions of three dimensional incompressible N-S equations are unique or not. claymath.org/millennium/Navier-Stokes_Equations $\endgroup$
    – Orcun
    Commented Oct 19, 2013 at 21:20

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