2d collisions

In my text, it says that the conservation laws alone do not suffice to predict the motions of two particles if the collision occurs in two dimensions. It is generally required that we know one of the final quantities in order to predict the rest. Does this mean that it is impossible to predict the outcome of a two dimensional collision using only the initial quantities? That is, is there some other unknown quantity (ie, other than velocity and mass) that would allow one to predict the outcome?
 

mathman

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There is one degree of freedom. Imagine a collision between two billiard balls on a pool table. The direction after collision is determined by the exact point of impact. On a microscopic scale, there cannot be an exact point of impact, so the direction after collision is based on a probability distribution.
 
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The initial conditions determine the final state, it's just that to figure out what that final state is, you have to look at the shapes of the objects, the details of the contact forces between them, etc.
 
Originally posted by mathman
There is one degree of freedom. Imagine a collision between two billiard balls on a pool table. The direction after collision is determined by the exact point of impact. On a microscopic scale, there cannot be an exact point of impact, so the direction after collision is based on a probability distribution.
Excellent! That's exactly what I suspected. I tried to solve the problem under the assumption that I knew the exact point of impact and I got a very complicated result (it was very tedious and I gave up after finding one of the final components of one of the masses). But I suspected that the unknown quantity was the direction of the impulse during the collision.
By the way, what exactly is a degree of freedom?
My astronomy professor said once, "The three-body problem is unsolvable because there are too many degrees of freedom" (or hopefully something close to that). This "unsolvable" problem also bothers me, because nature ought to be predictable. What makes the three body problem so difficult? Is it similar to the 2D collision problem in that some unknown cannot be known precisely?
 
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The Newtonian 3-body problem has a unique solution -- it just doesn't have a closed-form solution (i.e., a simple formula in terms of algebraic functions). That means in practice you have to resort to numerical approximation, although you can make the approximation as good as you like.
 
Originally posted by Ambitwistor
The Newtonian 3-body problem has a unique solution -- it just doesn't have a closed-form solution (i.e., a simple formula in terms of algebraic functions). That means in practice you have to resort to numerical approximation, although you can make the approximation as good as you like.
Would you say that this is due to a weakness of some sort in the manner of our mathematics? That is, is this problem difficult more due to the mathematics involved or more due to the physics? I have no idea what the solution looks like. It's just that I believe strongly in the power of math and it disappoints me when "numerical methods" have to be used.
 
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Originally posted by StephenPrivitera
Would you say that this is due to a weakness of some sort in the manner of our mathematics? That is, is this problem difficult more due to the mathematics involved or more due to the physics?
Neither. It's just that many -- in fact, most -- functions do not have closed-form formulas. For instance, many functions can only be expressed algebraically as infinite series. The 3-body problem happens to be an example of an equation whose general solution is not closed-form. There are infinitely many such equations; this one is just famous because a lot of people were interested in it. It's not a limitation of mathematics or anything; if anything, it's a limitation in what functions we consider to be "nice" (i.e., ones that we can write down simple formulas for).
 

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