A few days ago I came across a rather provocative paper discussing a surprisingly simple example of what the author considers to be indeterminism in Newtonian mechanics. Here's a link to the original paper if you'd like to check it out.

Here's the thrust of the argument:

Consider a point mass resting at the apex of a specially constructed frictionless dome, centered at r=0, z=0. It is azimuthally symmetric, and the height of the dome at distance r is z(r) = - (2 / 3g) r^(3/2). The leading coefficients are such that for m = 1, Newton's laws give us that r'' = r^(1/2). Given the initial conditions r = 0, r' = 0, there is an infinite family of solutions to this equation! This family is

r(t) = (1/144) (t - T)^4 for t >= T, r(t) = 0 for t <= T.

That is, there are solutions such that the mass will sit at the top of the dome up until time T, and then start spontaneously rolling down the dome at some unspecified angle phi.

Just to show these are solutions:

r'(t) = (1/36) (t - T)^3 for t >= T, r(t) = 0 for t <= T,

r''(t) = (1/12) (t - T)^2 for t >= T, r(t) = 0 for t <= T,

r''(t)^2 = (1/144) (t - T)^4 = r(t).

Newtonian mechanics gives us no indication which of these solutions to pursue, and is thus indeterministic.

I'm not going to get into many of the discussions the author has in his paper, but I will point out that he explicitly admits that this example isn't physically testable for a variety of reasons. His concern is mostly about what the theory itself says. I'm still chewing on this myself; what do others think of it?

LOL, I actually read through the author's example. Everything is so ridiculously wrong, this guy probably failed high school physics. Even without looking at his "calculation" you can see that he must be wrong - if the point mass sits at the top of the dome, it will stay there forever unless acted upon by a force with a component perpendicular to the symmetry axis of the cone. Such a force would necessarily break the symmetry of the problem, so there is no problem with indeterminism.

As for the "calculation" itself, the guy happily equates quantities that do not have the correct dimensions - first, the height of the cone h and the radius r must both have dimensions of length, but h = 2/3 g r^3/2 with some abitrary constant g. The author then says correctly that only a force with a radial component could set the mass in motion. For this, he chooses the gravitational force. Of course, either the gravitational force will act along the symmetry axis of the cone and have no radial component, so F = 0, or the force will break the symmetry, accelerating the mass in the proper direction, leaving no problem with determinism. He says the force acting on the mass will be F = g sin theta, but theta = 0, so again F = 0. Now, the g that was previously related to the height of the cone is suddenly the Earth's acceleration constant, and he cancels g on both sides The resulting equation of motion is d^2r/dr^2 = sqrt(r), which is also obviously wrong for dimensional reasons. The rest of the text is clearly irrelevant.

Cliffs notes: Move along, nothing to see here. Whenever you see a text that claims to have discovered a fundamental problem in physics but is written in MS Word using only 8th grade math, you should be very suspicious

Yes, obviously there is no problem if you assume the existence of an extra force. The reasoning that a force is necessary is that usually without an applied force the equations of motion will not allow for a solution of the form that he provides. I haven't sat down and done a bunch of calculations, but I'll be surprised if you can furnish me a "high school physics" example that starts with no velocity and no applied force and has valid solutions of this type. For example, let's take a simple harmonic oscillator, x'' = - w^2 x, x(0) = 0, x'(0) = 0. The solutions to this equation are x = A sin (wt) + B cos (wt), but to satisfy the boundary conditions, we get A = 0 (from the requirement that x'(0) = 0) and B = 0 (from the requirement that x(0) = 0). Thus the only solution to this equation is of the form x = 0 for all time.

I don't think your arguments about dimensional analysis have much merit, as you can fix the dimensions with a constant out front. Let C be some dimensionful constant such that r'' = C r^1/2 has units of acceleration (so C has units of r^(1/2) / s^2). Then r = C^2 / 144 (t - T)^4 still works, and as you can see it has the correct units. So this doesn't make the problem go away. If you want to get snarky in a more meaningful manner, you could point out that he is looking at the tangential component of the force (which has both radial and vertical components) but only considering radial acceleration. That is something I hadn't noticed before and might be important, but I need to do a bit of calculation before I know for sure.

Isn't this just an unstable equilibrium? So then what's the big deal? This is so 18th century.

The key to the dome is that it is a carefully chosen potential that violates a Lipschitz condition; as a result of this, some theorem about ODEs says that we no longer have unique solutions.

To show that this is different from any old unstable equilibrium, let's consider another case of unstable equilibrium - an ideal point pendulum standing straight up. The equation of motion is theta'' = - w^2 sin theta; for the boundary condition we're interested in, I'll make the substitution theta = pi + phi. Then phi'' = w^2 sin phi, and we're interested in the boundary condition where phi(0) = 0, phi'(0) = 0. Now, I'm going to do the same trick as normal and assume small phi so that we can write this as phi'' = w^2 phi. This will be justified when I demonstrate that phi = 0 for all t (which is clearly small.) The solutions to the approximate equation of motion are phi = A e^(wt) + B e^(-wt). phi(0) = 0 requires A = -B; phi'(0) = 0 requires A = B. The two constraints together demand the unique solution phi = 0 for all time. Even though this system is in unstable equilibrium, it still exhibits a unique solution to the differential equations. It will remain in unstable equilibrium indefinitely in this idealized example. This is not the same behavior as the dome problem.

Given that in the free will thread you seemed to be arguing that it is possible that physical law may prove indeterministic in a way that would allow free will, I'm amused that you're rejecting out of hand a proposal that actually would allow for indeterminism. I thought you might be interested in this.

The guy's whole point seems to be arguing that "Law 1. Every body continues in its state of rest ... unless it is compelled to change that state by forces impressed upon it." doesn't actually mean that a force must be applied/have been applied to cause something stationary to start moving.

Okay. Differential equations are not my thing, and I don't even know what the Lipschitz condition is. So I'll take your word for it that there's something interesting here. I'd be surprised if it is particularly novel though.

As far as free will is concerned, I'd say I vaguely believe it exists, but couldn't be bothered arguing that hard for it. Also, I had made the statement that I don't think that people can conclude that free will doesn't exist based on current knowledge of physics. You'd have to ask them if this example made any difference to them. I don't feel like I need this example to make that point. In any case, Newton's laws don't describe reality exactly.

the conclusion of the article appears correct, in a sense Newtonian mechanics isn't deterministic. while this is pretty cool, it doesn't really have any practical or philosophical implications.

Sorry, but you missed my point. This guy does NOT give an example of a point mass suddenly accelerating without a force acting on it. There is an applied force, namely gravity, but there are no valid solutions of the type he provides. This is just an example of Garbage In, Garbage Out, since the equation of motion he "solves" is wrong from the start. The dimensional analysis is just one additional thing that is wrong. The fudge factor c that you want to introduce has just the dimensions he lost by equating the g related to the dome height (dimensions length^-2) to the completely unrelated quantity g' that is the Earth's acceleration (dimensions length/time^2).

The crucial point is that the radial component of the gravitational force, which is the only component that will accelerate the mass in the radial direction, is simply zero. By a sleight of hand, namely by using the symbol r with two different meanings. He is confusing the symbol for the function r(t) with the value of the argument of the dome height h, h(r = 0), and so he arrives at the wrong equation of motion, r'' = r^1/2. The correct equation of motion is r'' = 0.

He is basically trying to sell to you the idea that the equation r''(t) = 0 with r(0) = 0 and r'(0) = 0 has non-trivial solutions of the kind you mentioned. If you believe that, then I cannot help you anymore. As others have said, this is just an example of an unstable equilibrium, nothing else.

...

The equation does NOT correctly represent the situation. So much for philosphy guys trying to do a free body diagram. The cooridnate system in use does not work.

This is true. However, the normal force associated with the dome itself does have a radial component. So this . . .

is false. Unless your high school physics class taught you that balls can't roll downhill; this is exactly the same analysis that's associated with a ball rolling down an inclined plane.

EDIT:

This appears to be true, since he is using a tangential force to consider a radial acceleration, which is sloppy to be sure. If we consider the system as constrained to lie on the dome surface - so that the relationship between z and r is assumed to be held fixed - we can write down a Lagrangian for the system and get an equation of motion that way, and it does look pretty ugly. I'll spend a little time today playing around to see if you can fix this. If you can't, that might be pretty interesting.

EDIT 2: However, I'll also say this. What the philosopher is trying to do is come up with a physically plausible sounding case that violates the uniqueness condition for ODEs. But, even he'll admit that it's impossible to set up the exact situation that he describes in his paper. I think his main concern is that there is a possibility for indeterminism in the theory just based on the possible non-uniqueness of the ODEs, and there's nothing a priori in the theory that says that you can't just pick a 1-D potential that violates those conditions. If no such potential can be found in real life, then that's a statement of real physical content apart from the theory.

I think the author identifies the problem himself, but he somehow fails to connect the dots. In Section 2.3, he writes,

So the right way, according to Newton, to determine the motion of an object is to apply Newton's laws in discrete time to obtain a polygonal path of motion, and then take a limit as the segment size goes to zero. So let's do that for the dome.

In this problem, Newton's laws give us the following difference equation:

r(t + ∆t) = r(t) + v(t) ∆t,
v(t + ∆t) = v(t) + (r(t))^{1/2} ∆t,
r(0) = v(0) = 0.

If we define h = ∆t and t_n = nh, then this becomes

(1) r(t_{n+1}) = r(t_n) + v(t_n)h,
(2) v(t_{n+1}) = v(t_n) + (r(t_n))^{1/2}h,
(3) r(t_0) = v(t_0) = 0.

By solving (1) for v(t_n) and substituting the result into (2), we can combine this into a single difference equation:

(4) (r(t_{n+2}) - 2r(t_{n+1}) + r(t_n))/h² = (r(t_n))^{1/2},
(5) r(t_0) = r(t_1) = 0.

According to Newton, we are supposed to solve this difference equation to get a polygonal path of motion, and then take the limit of that path as h goes to zero. Well, one of the nice features of difference equations like this is that they have unique solutions. The value at any time is obviously completely determined by the previous values. In this case, the unique solution is the zero solution: r(t_n) = 0 for all n. Now we take the limit, and the limit of zero is zero. So the path of the ball on the dome, according to Newton, will be r(t) = 0 for all t.

So what's the problem? The problem is that, in practice, we rarely solve problems this way, because difference equations can be notoriously difficult to solve. Instead, we try to employ a mathematical trick. Newton says we should solve (the difference equation) and then take the limit. But rather than do solve-limit, we usually try to do limit-solve. In this problem, we would take the limit of (4), which gives

(6) r''(t) = (r(t))^{1/2}.

We hope that when we solve this, we end up at the same place that Newton's recipe is supposed to take us. But the trick does not always work. We have precise mathematical theorems that tell us when this trick will work. The dome is an example where the trick does not work, because the limiting ODE does not have a unique solution.

In the end, the only thing interesting about this example is that it illustrates a principle well-known to every mathematician, but probably very few laypeople: the limit of a (solution/integral/etc.) is not always the (solution/integral/etc.) of the limit.

Nonsense. It tells us precisely which solution is correct: r(t) = 0. It tells us this because r(t) = 0 is the unique limit of the unique solution to the unique difference equation that is completely determined by Newton's laws. All of those other "solutions" are red herrings that arise when a mathematician tries to "cheat" Newton by reversing the order of operations.

Here are some additional comments.

1. The mathematical typesetting looks atrocious, and makes the paper look amateurish.
2. It looks like the author has made a little cottage industry out of the dome. He has follow-up papers in which he argues that the dome example proves the uselessness of Bayesian inference. He then has other papers where he describes his own systems of inference that should replace Bayesianism when it is cast aside.
3. The author has a shorter version of the paper, which he says is "prepared for publication." In the short version, the whole discussion of polygonal paths has been removed. Apparently, he does not feel this is worth publishing, despite his acknowledgment that it is right there in Newton's original formulations. Perhaps he realizes on some level that this is the key to unraveling his whole "paradox".

Now that is the kind of post I was trying to get out of this thread. Thanks, jason, that was definitely illuminating. EDIT: And yes, it is a rather ugly looking paper.

In other words, your math is correct but your physics is appalling.