By Olivier Vallée

Using designated capabilities, and specifically ethereal features, is quite universal in physics. the explanation could be present in the necessity, or even within the necessity, to precise a actual phenomenon by way of an efficient and finished analytical shape for the full clinical group. even though, for the previous two decades, many actual difficulties were resolved via desktops. This pattern is now turning into the norm because the value of desktops keeps to develop. As a final hotel, the designated services hired in physics must be calculated numerically, whether the analytic formula of physics is of fundamental value.

Airy capabilities have periodically been the topic of many overview articles, yet no noteworthy compilation in this topic has been released because the Nineteen Fifties. during this paintings, we offer an exhaustive compilation of the present wisdom at the analytical houses of ethereal capabilities, constructing with care the calculus implying the ethereal capabilities.

The publication is split into 2 elements: the 1st is dedicated to the mathematical houses of ethereal capabilities, while the second one provides a few functions of ethereal features to numerous fields of physics. The examples supplied succinctly illustrate using ethereal services in classical and quantum physics.

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**Example text**

3 (Arnold, Margulis, Pyartli). If some real-analytic map s → ωso from a domain of R p to Rm is non-planar in the sense that its image is nowhere locally contained in some proper vector space of Rm , the Lebesgue measure of {s, ωso ∈ HDγ ,τ } is positive provided that γ is small enough and τ large enough. 1 (for the spatial (resp. planar) secular system p = 2n (resp. p = n)). Hε (λ , Λ , Z) = H 0 (Λ ) + ε Hε1 (Λ , Z) + ε Hε2 (λ , Λ , Z), with Hε1 (Λ , Z) = h0 (Λ ) + ∑1 2 j 2n τ j (Λ )|Z j | + 0(|Z|4 ), and Hε2 has zero average with respect to λ ∈ Tn .

For any T > 0, any local minimizer of the action among paths x(t) = (r1 (t), r2 (t), · · · , rn (t)) in the configuration space which start at x(0) = x and end at x(T ) = x is collision-free, and hence a true solution of Newton’s equations, in the open interval ]0, T [. Already in the case of two bodies, this theorem is non-trivial. Translated in terms of the Kepler problem, it asserts that given two points x , x ∈ R2 \ 0 and T > 0, a minimizing path x(t) ∈ R2 \ 0 x(0) = x , x(T ) = x , is a collision-free solution of the equation x(t) ¨ = −x/||x(t)||3 .

This is possible if Λ belongs to the set A1 on which ν (Λ ) ∈ HDγ ,τ . Moreover, Whitney regularity allows to extend this to a (non unique) symplectic transformation L such that Hε ◦ L keeps the same form with Hε2 (λ , Λ , Z)) replaced by R1 (ε , λ , Λ , Z) + O(ε N1 ), where R1 vanishes at infinite order along {(ε , λ , Λ , Z)|Λ ∈ A1 }. 2. Transformation to Birkhoff normal form up to order N2 . This is possible if Λ belongs to the subset A2 of A1 defined by diophantine conditions on the set (ν1 , .