# Determinism of classical mechanics

1. Feb 15, 2012

### zezima1

Is classical mechanics deterministic?

Suppose we collide two bodies with each other. Assuming they are point particles and using conservation of energy and momentum this gives us a set of equations. Unfortunately these aren't enough to predict their trajectories. To do this we will also need the angle between the trajectories after the collision. Now that doesn't sound very deterministic.
But so I thought that imagining bodies as point particles is not really a valid thing to do in classical mechanics, and that might be the reason why the problem is not deterministic.

So instead I tried to scale them up to an everyday size, i.e. two hockey pucks. Treating them as rigid bodies you can use conservation of angular momentum etc. to obtain even more equations. Problem is however that you still get too many variables.

So indeed, how can you say that clasiccal mechanics is deterministic following this example.

2. Feb 15, 2012

### Drakkith

Staff Emeritus
The key is that IF you can know all the variables and such you can determine the outcome of anything in Classical Physics. Without knowing everything you cannot determine anything down to the absolute smallest detail. It isn't actually possible in real life as there is simply no way to know all the variables of a system. Not to mention Quantum Mechanics has shown us that nature isn't deterministic. You simply cannot know the exact conditions of any system, ever.

3. Feb 15, 2012

### Ken G

What's more, you have put your finger on the original objection people had to Newton's approach. Newton said that to know everything about the future, you have to know everything about the present. This raises two problems that people didn't like:
1) they wanted to know why things are the way they are, but they couldn't know that without knowing why things were the way they were, and
2) there is no way to know everything about the present, so there is no way to know everything about the future.
Problem #2 speaks to your issue above. But despite these two significant objections, it was found all the same that Newton's approach does tell you what you need to know (what will happen if I do X given Y), if not what you want to know (why Y). So we kind of stopped objecting to it! (But we should not have been so surprised later on when it was found that Newton's approach was incomplete.)

4. Feb 15, 2012

### aaaa202

Know all the variables? Deterministic means that you can predict what happens from knowing all the initial variables.

... And what I wrote was indeed under the assumption that you know all about the initial conditions!

5. Feb 15, 2012

### Ken G

So is your question that it still seems to you, even if you know all the initial conditions, that you'd have too many variables in the final condition? That can't be true, since the final condition involves all the same variables as the initial condition, so there must be the same number of them. You are right that you need to know something about the nature of the collision, like is it hockey pucks and is it elastic, etc., but if you do know what kind of collision you are talking about, you don't need to specify the angle after the collision, you can figure that out given the initial conditions. Perhaps the key point is that you do have to say something specific about the nature of the interaction, otherwise I can just turn all the interactions off and I know perfectly well I can get a valid solution where everything just keeps doing what it was doing before.

To me, the key point in all this is that the concept of determinism never worked at all without also making certain idealizations. One must idealize the initial conditions as being exact, even though they never are, and one must idealize the nature of the collision as being exactly known, even though it never is. Hence, the entire notion of determinism in classical physics was never anything beyond an idealization, and as such should not be taken as seriously as it is, even without ever mentioning quantum mechanics. In practice, any concept of determinism must account for uncertainties in the initial data and uncertainties in the interaction physics, so once you account for all these uncertainties, all you have left is a form of statistical determinism that is actually quite similar to quantum mechanics uncertainties except that the latter are generally much smaller (not larger!) than the former. The idea that the uncertainties in classical determinism could in principle be made arbitrarily small was never anything but an unsubstantiated leap of faith.

Last edited: Feb 15, 2012
6. Feb 15, 2012

### zezima1

Try it for yourself. Assume that two particles with momentum (p1x,p1y) and (p2x,p2y) with masses m1 and m2 collide in a completely elastic collision.. I want to see an expression for the angle.

7. Feb 15, 2012

### AlephZero

Work in the inertial coordinate system such that the CM of the two particles is at the origin before the collision. Since the forces are all internal, the CM remains at the origin after the collision.

In this coordinate system there is no "angle". The particles approach along a straight line and separate along the same straight line.

Then transform back into your original coordinate system to find your "angle".

8. Feb 15, 2012

### lugita15

I think what zezima1 wants, among other things, is a proof of the following statement: if the center of mass of particles A and B is stationary, and particles A and B have an elastic collision in a straight line, then they will separate along the same line.

9. Feb 15, 2012

### Andy Resnick

Do you admit statistical mechanics as part of 'classical mechanics'?

10. Feb 15, 2012

### Pythagorean

As I think has been pointed out, unknown variables doesn't weigh in on determinism.

We generally model classical systems deterministically (i.e. their phase trajectories don't cross in phasespace so that there's only one future for each point in the state space). But then some people will add a small noise term, making their system mathematically non-deterministic, but not really modern physics either.

Of course, one could still argue that this noise term is deterministic, but we model it randomly because it's simpler and yields the same qualitative results.

So it always becomes a philosophical issue in the end about how you interpret the discrepancies between model and observation.

11. Feb 15, 2012

### Drakkith

Staff Emeritus
I don't believe there is enough information to determine this. You haven't given us any way to determine when the particles impact each other or where. So we don't actually fully know the initial conditions for the system.

12. Feb 15, 2012

### Pythagorean

I'd prefer they weren't point particle; let's do spherical dirt clods. I'd also like to see compression, friction, hydrogen bond breaking, and thermodynamics of the collision in this problem. So really it's a collision of two ensembles. Thanks for your hard work.

(i honestly don't think it would be deterministic anymore if you considered all of this)

13. Feb 15, 2012

### AlephZero

Clearly a rigid particle with zero size is a mathematical abstraction, not something "real". I think Newton would have answered that question with "Hypotheses non fingo". Or as the first "rule of reasoning in natural philosophy" puts it in Principia, "we are to admit no more causes of natural things that such are are both true and sufficient to explain their appearances".

Another way to look at it is using symmetry. Why should there be a collision force that causes a sideways deviation in one direction, rather than the other direction? A Newtonian point particle doesn't have any properties that can break the symmetry here.

For collisions between finite sized bodies, there is a mechanism for generating a sideways force, namely friction. When you specify a model of the frictional forces (e.g. Coulomb's "laws" of friction) you have sufficient information to get a unique solution. But I don't see the purpose of inventing a hypothetical effect in collisions between point particles, for the sole purpose of making the situation indeterminate, unless you can provide some experimental evidence to back up your assertion that it is indeterminate.

14. Feb 15, 2012

### lugita15

I think the only point of even thinking about this possibility is to demonstrate that Newtonian mechanics for point particles does not by itself logically force determinism (although with very weak additional assumptions, like the symmetry considerations you brought up, it does very clearly lead to complete determinism).

15. Feb 15, 2012

### Rap

There's no such thing as a point particle in classical mechanics. There are times when you can pretend an object is a point particle and get a very good answer, other times not. For example, if you have two solid balls gravitationally attracting each other, and their radii are much, much smaller than the distance between them, you can pretend they are point particles and get a very good answer. But if they collide, you break your original assumption that they are far apart compared to their radii, and then you cannot treat them as points. Now they are solid balls, and you have to know the details of the collision in terms of balls, not point particles. Any time you are pretending you have point particles in classical mechanics and come up with such an indeterminacy, you have violated the assumptions you made that allowed you to treat them as point particles.

16. Feb 15, 2012

### lugita15

What law in classical mechanics forbids point particles?

17. Feb 16, 2012

### Rap

None. Just because they are not forbidden does not mean they exist. What experiment proves they exist?

18. Feb 16, 2012

### lugita15

Well, classical mechanics has been superseded anyway by later theories, so the whole question of whether or not point particles could or would exist classically is a purely theoretical issue, not one amenable to experimental testing. And I maintain that point particles make total sense classically.

19. Feb 16, 2012

### Pythagorean

If you find yourself getting too comfortable with point particles, just imagine what the force of a negative charged particle on a positive charged particle right next to it. Since they're two point particles and they touch, d = 0. Thus, the electromagnetic force (or gravitational force) between them is infinite.

20. Feb 16, 2012

### Rap

They make sense only under certain conditions. When those conditions are not met, they make no sense. The collision of two classical point particles is a situation where those conditions are NEVER met.

Classical physics NEVER uses the concept of a point particle except as a limiting condition. "In the limit of blah blah" and if "blah blah" is not true, the object cannot be approximated as a classical point particle. "blah blah", whatever it is, is NEVER true when considering the collision of two classical bodies. You cannot give one instance where classical physics deals with a point particle except as an approximation.

Your position is that point particles make total sense classically. How do you defend this position? What is your argument? If it is simply an opinion, then what you are saying is that determinism in classical physics is less fundamental than your opinion, and I can't believe you mean that.

Why not adopt the viewpoint that the lack of determinism in the collision of classical point particles is proof that they do not make sense under all classical conditions?