Indeterminism in Classical Physics

[klen]

I was reading the Roger Penrose book Emperor's New Mind and he was explaining the determinism in Newtonian mechanics.
He says that if we consider two solid balls colliding (assuming elastic collision) then outcome depends continuously on initial state of the balls.
But if we consider triple or higher order collisions say three balls A,B,C come together at once it makes a difference if we consider A and B come together and then C to collide with B immediately afterwards or if we consider A and C to come together and then B to collide with A immediately afterwards.
From this he concludes that there is indeterminism in exact triple collisions and the output depends discontinuously on the input state.
I don't quite understand above conclusions. Could anyone please explain it to me??

 

[Jano L.]

First important thing is to define determinism. The word is used in various shades of meaning.

The definition I like is this: determinism is a property of mathematical model. The model is deterministic if the state of the model system at one time is sufficient to determine its state at all past and future times.

The motions of perfect balls are rectilinear except when they collide; in collision, they change their velocity.

It is true that slight variation of the initial condition can lead to great variation of the resulting motion. In case of perfectly solid spheres, the collisions are instantaneous. This allows for the possibility that the resulting motions after the collisions are discontinuous functions of the initial conditions.

The initial condition for which there is discontinuity in the motion is special, because due to the discontinuity, the motion for this condition is not determined. It can be anyone from those defining the discontinuity. Hence the model is not entirely deterministic.

 

[klen]

Jano thanks for your reply it was helpful

 

[lightarrow]

Well, talking of ideal bodies in Newtonian mechanics, it's not determined even the problem of finding the constraint reactions of a four-feet rigid body resting on the pavement.

 

[PJC01]

I would respond to this content by acknowledging that there is indeed a concept of indeterminism in classical physics. This refers to the idea that the future state of a system cannot be determined solely by its present state and the laws of physics. This is in contrast to determinism, which states that the present state of a system can uniquely determine its future state.

In the context of the example given by Penrose, he is highlighting the fact that in certain scenarios, such as triple collisions, the outcome may not be completely determined by the initial state of the system. This is because the order in which the collisions occur can affect the final outcome. This is known as the "butterfly effect" in chaos theory, where small changes in initial conditions can lead to vastly different outcomes.

Penrose's conclusion about the discontinuous dependence of the output on the input state in triple collisions is based on the fact that the slight change in the order of collisions can lead to a significantly different outcome. This highlights the limitations of determinism in predicting the behavior of complex systems, and the presence of indeterminism in classical physics.

It is important to note that this concept of indeterminism does not necessarily mean that the laws of physics are not accurate or reliable. It simply means that there are certain scenarios where the outcome cannot be determined with absolute certainty. As scientists, we continue to explore and understand the complexities of the physical world, including the concept of indeterminism, in order to further our knowledge and understanding of the universe.