"Three billiard balls collide simultaneously" revisited after 20 years

[Carl K]

Greetings,

I used a new perfect (but simple) Newtonian physics engine to reverse a
billiards break. The simulation surprised me. I expected it to re-form a
stationary triangle of balls and spit out the cue ball. It did not.

This means that classical physics is non-deterministic and not practically
reversible.

The simulator is perfect in that it keeps times, positions, and velocities
as expressions--3*sqrt(7)/7--rather than using numeric approximations. Its
worlds are 2D with circles, infinite walls, and elastic collisions.

When trying to understand the results, I found a useful thread from
sci.physics.research from almost 20 years ago. I eventually also found a
few references in the philosophy of science.

If you're interested in classical physics, determinism, or reversibility,
see the details, short videos, and open-source code here (free):
https://towardsdatascience.com/perfect-infinite-precision-game-physics-in-python-part-2-360cc445a197
The old sci.physics.research thread:
https://groups.google.com/g/sci.physics.research/c/xOCFItVlnro?pli=1

- Carl

[Phillip Helbig (undress to reply)]

Post by Carl K
I used a new perfect (but simple) Newtonian physics engine to reverse a
billiards break. The simulation surprised me. I expected it to re-form a
stationary triangle of balls and spit out the cue ball. It did not.
This means that classical physics is non-deterministic and not practically
reversible.

It does not mean that it is non-deterministic. Make a movie of the
billiard break and run it backwards: perfectly valid physics. Classical
physics is reversible.

That it is not reversible in practice is a different matter, but that
was known before the simulation.

Post by Carl K
The simulator is perfect in that it keeps times, positions, and velocities
as expressions--3*sqrt(7)/7--rather than using numeric approximations. Its
worlds are 2D with circles, infinite walls, and elastic collisions.

But even 3*sqrt(7)/7 or whatever, when it comes down to actually
calculating something, will be used a finite approximation.

Taking a broader view, questions such as whether classical physics is
deterministic and/or reversible aren't determined by simulations. There
are some complicated processes which simulations can shed light on, but
with regard to fundamental issues, one must know the fundamental issues
when programming the simulation.

[Michael Moroney]

Post by Phillip Helbig (undress to reply)

Post by Carl K
I used a new perfect (but simple) Newtonian physics engine to reverse a
billiards break. The simulation surprised me. I expected it to re-form a
stationary triangle of balls and spit out the cue ball. It did not.
This means that classical physics is non-deterministic and not practically
reversible.

It does not mean that it is non-deterministic. Make a movie of the
billiard break and run it backwards: perfectly valid physics. Classical
physics is reversible.
That it is not reversible in practice is a different matter, but that
was known before the simulation.

Is it chaotic in a way, that is, a tiny change in the backwards action
causes a completely different outcome?

[Phillip Helbig (undress to reply)]

Post by Michael Moroney

Post by Phillip Helbig (undress to reply)

I used a new perfect (but simple) Newtonian physics engine to revers=

e a

Post by Michael Moroney

Post by Phillip Helbig (undress to reply)

billiards break. The simulation surprised me. I expected it to re-fo=

rm a

Post by Michael Moroney

Post by Phillip Helbig (undress to reply)

stationary triangle of balls and spit out the cue ball. It did not.
This means that classical physics is non-deterministic and not pract=

ically

Post by Michael Moroney

Post by Phillip Helbig (undress to reply)

reversible.

It does not mean that it is non-deterministic. Make a movie of the
billiard break and run it backwards: perfectly valid physics. Classi=

cal

Post by Michael Moroney

Post by Phillip Helbig (undress to reply)
physics is reversible.
That it is not reversible in practice is a different matter, but that
was known before the simulation.

Is it chaotic in a way, that is, a tiny change in the backwards action
causes a completely different outcome?

Right. Even if one could specify the positions and velocities as well
as allowed by the uncertainty principle, after just twelve collisions
the uncertainty in a ball's position would be larger than the size of
the table.

[Carl K]

Summary:

I'm the original poster. I recently found a reference with a simpler
example illustrating the issue.
Imagine this simple Newtonian example from Lee [2016,
https://dl.acm.org/doi/pdf/10.1145/2912149]: Three balls of unequal mass
in the line; the left is moving right, the middle ball is stationary; the
right ball is moving left. This seems to show different results depending
on whether the left ball hits first or the right ball hits first. Prof.
Lee concludes that the Newtonian system is mathematically "incomplete" in
the sense that it does not include its limits (Here, the limit as the time
delta between the hits goes to zero).

Details:

I'm interested in the mathematical properties of Newtonian physics and a
few days ago found this paper by Prof. Edward A. Lee of U. of California,
Berkeley. Fundamental Limits of Cyber-Physical Systems Modeling, November
2016, ACM Transactions on Cyber-Physical Systems 1(1):1-26

He gives the example above in section 4, on page 17. I worked it out with
his suggested masses of 1/5, 1, and 5 and before velocities 1, 0, -1 and,
as he said, got different sets of after velocities. Specifically, I used
my simple physics engine and got <-73/27, -40/27, -5/9> vs <-31/9, -7/9,
-2/3>.

He closes the paper with a discussion of issues such as interleaving,
butterfly effect, superposition, and determinism. If interested, please
see the paper (starting near page 18) for more details.

His lab works on real-world physical systems, so he also discusses more
realistic models (for example, one with springs).

I personally love his example. I find it much more compelling than other
similar attempts to understand the mathematical structure of Newtonian
physics (e.g. Norton's dome, Suppes' 2D symmetric 3-particle collision,
and various 2D "game engine" collisions.)

- Carl
https://medium.com/@carlmkadie