By H. Goldstein
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Additional info for Classical Mechanics
3 State-vector In the Langevin treatment of the Brownian motion (Langevin, 1908), the relevant variables to describe the whole physics are the velocity and the position: x, v. These two variables deﬁne therefore the state-vector of the system. Of course other choices can be possible, for instance the sole position as it was done by Einstein, or adding other variables like acceleration. It appears that a hierarchy between state-vectors naturally arises with regard to the information content desired in the physical approach (Minier and Peirano, 2001): the more information is detailed, the larger the number of variables possibly contained in the state vector.
Such a step not only allows researchers to be free from any misconceptions (whereas even distinguished physicists fell into the poor belief that Ito deﬁnition was ‘a mathematical vagary’. . ) but, more importantly, will avoid them to induce spurious drifts by ﬂawed mathematical calculus and also potentially inconsistent numerical schemes. Bibliography L. Arnold, Stochastic Diﬀerential Equations, theory and applications, John Wiley & Sons, New-York, 1974. I. Karatzas and S. E. Shreve, Brownian motion and stochastic calculus, Springer Verlag, Berlin, 2cd edition, 1991.
The existence of the extra term entering Ito calculus compared to classical calculus is actually simple to understand: it stems from a Taylorseries development made to the second-order in dt since it must be remembered that (dWt )2 = dt in a mean-square sense and, therefore (contrary to the rules of classical calculus with diﬀerentiable functions), second-order terms of the development can give contributions to the ﬁrst order √ in dt. As a practical rule-of-the-thumb, it is often written that dWt ∼ dt.