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Extra resources for Beam Dynamics Issues for 1 TeV Linear Collider
Even in a fully deterministic world, the assumption of a Laplacean demon which can calculate the Newtonian universe in the long run will eventually be exposed as an illusory fiction. 3 Hamilton Systems and the Chaos of Heaven and the Quantum World In the 18th and 19th centuries, Newtonian mechanics seemed to reveal an eternal order of nature. From a modern point of view, Newtonian systems are only a useful kind of dynamical system for modeling reality. In order to specify the initial state of a Newtonian system, the positions and the velocities of all its particles must be known.
With period 2rr). Mathematically, we assume J continuous with Iimited variation. 4) n=-oo Therefore it can easily be proved mathematically that J(t) can be approximated by means ofsums . 5) n=-N with any desired degree of exactitude for increasing N. Function f is indeed uniformly convergent. 6) Astronomically, this result means that a constant-motion path (oflimited variation) can be approximated to any desired degree of exactitude by means offinite superpositions of the epicyc1e motions. Is it c1ear that so far we have used only superpositions with epicyc1e periods ±2rr, ±rr, ±~rr, ±~rr, ±~rr, ....
In the ca se of early astronomy and mechanics, this was the first step of mathematical idealization and led to a geometrie model for the set of idealized states which is nowadays called the state space of the model. The presocratic 'models' of nature differ from modern ones not only because of their mathematization and measurability, but also because the relationship between the actual states of a real system and the points of the geometrie model was believed to be ontologically necessary, while in modern systems it is a fiction maintained for the sake of theory, prediction, and so on.