Birkhoff Transformation

We remark that the singularities are now located at Pi(—0), P2(3,0). The aim of the Birkhoff Transformation will be to regularize simultaneously both collisions with Pi and P2. To this end, let us write the equations of motion as... 

In summary, the regularization steps performed to achieve Birkhoff Transformation are the following. Let... 

As a side step, we will briefly discuss two topics not directly related to our primary objective, perturbation theory. Both the inverse map and the Birkhoff Transformation in three dimensions allow for an elegant treatment in terms of quaternions. 

Here we will first revisit the classical Birkhoff Transformation (the same conformal map is known in aerodynamics as the Joukowsky transformation) and represent it as the composition of three conformal mappings this will then readily generalize to the spatial situation by means of quaternions. 

Figure 5. The sequence of conformal maps generating the Birkhoff transformation.

The anharmonicities of the potential contribute by the terms involving the constants x, g, y,. .. as well as the energy shifts AEx = 0(h2),. .. and the frequency shifts Aw, = 0(h2),. These anharmonic constants can be calculated by the Van Vleck contact transformations [20] as well as by a semi-classical method based on an h expansion around the equilibrium point [14], which confirms that the Dunham expansion (2.8) is a series in powers of h. Systematic methods have been developed to carry out the Van Vleck contact transformations, as in the algebraic quantization technique by Ezra and Fried [21]. It should be noted that the constants x and g can also be obtained from the classical-mechanical Birkhoff normal forms [22], The energy shifts AEx,... 

Until now, we have discussed NHIMs in general dynamical systems. In this section, we limit our argument to Hamiltonian systems and show how singular perturbation theory works. In particular, we discuss NHIMs in the context of reaction dynamics. First, we explain how NHIMs appear in conventional reaction theory. Then, we will show that Lie permrbation theory applied to the Hamiltonian near a saddle with index 1 acmally transforms the equation of motion near the saddle to the Fenichel normal form. This normal form can be considered as an extension of the Birkhoff normal form from stable fixed points to saddles with index 1 [2]. Finally, we discuss the transformation near saddles with index larger than 1. 

This regularizing transformation was proposed in 1915 by George David Birkhoff (Birkhoff 1915) in order to regularize all singularities of... 

The phase space structures near equilibria of this type exist independently of a specific coordinate system. However, in order to carry out specific calculations we will need to be able to express these phase space structures in coordinates. This is where Poincare-Birkhoff normal form theory is used. This is a well-known theory and has been the subject of many review papers and books, see, e.g.. Refs. [34-AOj. For our purposes it provides an algorithm whereby the phase space structures described in the previous section can be realized for a particular system by means of the normal form transformation which involves making a nonlinear symplectic change of variables. 

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