Magnus approximation

There is a close connection between the adiabatic frequency and the stability frequency (cf. Eq, 19) of a periodic orbit. Since at the pods the time average of is exactly d Em/du one finds that the adiabatic stability frequency is just the first order Magnus approximation to the exact stability frequency.From a practical point of view it is actually easier to compute the stability frequency. Finding the adiabatic frequency implies actual construction of the (u,v) coordinate system. 

From the properties of the Magnus approximation, one can show that if 3T<< than the adiabatic approximation is excellent. Thus, given a pods, one may easily compute its period and stability frequency and so assess the validity of the adiabatic approximation in the vicinity of the pods. 

The order of approximation which obtains may be seen from the following figures (A. Magnus, 1910) ... 

A. Magnus, in Pade Approximation and its Applications, Lecture Notes in Mathematics, 765, L. Wuytack, ed.. Springer, Berlin, 1979, p. 150. 

A. Magnus in Pade Approximation and its Applications, Lecture Notes in Mathematics 888,... 

Morphine s synthesis remains a serious challenge to this day. Until recently, the formal synthesis published by Kenner Rice [16] was its most efficient preparation. In 2009, Magnus reported a route to codeine with a reported overall yield of approximately 17% [17]. All academic s)mtheses reported in the literature, creative as these may be, suffer from lack of practicality, with the sole exception of Rice s disclosure, which has potential for scale-up. 

Alduchov OA, Eskridge E (1996) Improved Magnus form approximation of saturation vapor pressure. J Appl Meteorol 35 601-609... 

As mentioned in Section 2.5, optical excitation in the practically somewhat trivial case of the degenerate many-level problem with a constant diagonal matrix W can be treated by exact integration, because the Magnus expansion terminates after the first term. Closed expressions for the two-level (also the three-level) problem can thus be obtained, and can be compared, for instance, with the quasiresonant approximation, identical with the RWA for the two-level model. Since the conditions for the QRA are obviously not satisfied (x = co in this case and not x tu, as required by the QRA), there are then sizeable differences between the exact and the approximate results. ... 

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