Classical Uncertainty Predicting the Future

Newton s laws are perfectly deterministic. Suppose we have a set of masses, all of whose positions and momenta are specified at some instant in time. Further, suppose that we specify all of the forces of interaction between these balls. We could now use Newton s laws to determine, exactly, the state of this system at any later time. 

For example, suppose we knew that at time t = 0, a ball was centered in a box with length 0.5 meter, and suppose we knew that it was moving to the right at exactly 1ms-1. In the absence of any forces, we could predict exactly where it would be 

For short times we know the approximate position, but at long times the position is essentially unknown. However, there is no fundamental physical limit to our ability to measure the velocity so, in principle, we can do the initial measurement with enough accuracy to predict the position at any later time. For example, we could accurately predict the position of the ball after one year (31,557,600 seconds) if we measured the initial velocity with an uncertainty much better than one meter in 31,557,600 seconds. 

According to Newtonian physics, the universe and all of the objects in it are simply an extremely complicated collection of masses and charges. This implies that if you knew the state of the universe at any one time (exactly), you could predict (exactly) the future. Of course it is unrealistic to actually do a set of measurements which define the conditions at one time with enough accuracy, but it is in principle possible. Therefore, Newtonian mechanics predicts that the future is perfectly determined by the past. There is no such thing as free will , and you need not worry about studying for the next examination it has already been determined how much you will study, and what you will score on the test  

FIGURE 5.9 Uncertainty in position, coming from uncertainty in initial knowledge of velocity. This uncertainty can be reduced to an arbitrarily small value by a sufficiently accurate initial measurement. 

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