Birkhoff normal form

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,... 

Thus, based on NHIMs with saddles with index 1, we can construct a theory that is a rigorous reformulation of the conventional Transition State Theory [9,10]. Moreover, the use of the Lie perturbation brings the system locally into the Birkhoff normal form with one inverse harmonic potential [2]. This form is nothing but the Fenichel normal form. 

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. 

Note that, for those terms with oti = pj, the problem of eliminating higher-order terms is the same as that in the Birkhoff normal form for stable fixed points. Then, it is possible to eliminate higher-order terms as long as the set of frequencies (d n = 2,..., N) satisfies the nonresonance condition... 

The standard scheme of proof of the theorem is composed by a so called analytic part, based on the construction of a local Birkhoff normal form in suitably chosen domains, and of a geometric part which takes care of covering the phase space with good domains. A remarkable exception to this scheme is the paper by Lochak (1992), where the geometric part is replaced by a clever use of convexity and of the simultaneous approximations of real numbers with rationals. We shall sketch the traditional scheme. 

Some arbitrariness is still involved in the choice of the transverse Palais section V in Figure 3.4. This geometric arbitrariness can in fact be used to further simplify the skew product form (3.11). Such an approach, which first simplifies v = (f v) to Poincare-Birkhoff normal form, and then redefines V to further simplify the cross term a v), has been pursued in [23]. 

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. 

R. Perez-Marco, Convergence or generic divergence of the Birkhoff normal form, Ann. Math. 157 (2003) 557-574. 

The classical and most common attempt, already discussed by Poincare, consists in trying to remove all dependencies on the angles from the Hamiltonian. This is usually called the normal form of Birkhoff. Such a normal form turns out to be particularly useful when the unperturbed Hamiltonian is linear, i.e., in (1) we have Ho(p) = (u),p) with some lo R". Therefore we illustrate the theory in the latter case. However, with some caveat, the method is useful also in the general case see Section 4.3 on Nekhoroshev s theorem. 

The concept of complete stability is due to Birkhoff (1927), who introduced it in connection with the study of the normal form of a Hamiltonian system in the neighborhood of an equilibrium. However, the same method works properly also if one considers the neighborhood of an invariant torus, as we are going to discuss. We shall use a direct construction of the first integrals, which turns out to be quite simple. 

In such a domain we may perform the construction of Birkhoff s normal form illustrated in Section 2.5. To this end we split again the Hamiltonian as a series... 

Let us now proceed in a different manner, which however is essentially equivalent to the procedure illustrated there. Instead of looking directly for first integrals, we go through the process of constructing a Birkhoff s normal form. This is what we have done in Section 2.5, at least formally we have actually seen that we can construct a normal form up to an arbitrary large order. Thus, we may give the Hamiltonian the form... 

M. Kaluza, M. Robnik, Improved accuracy of the Birkhoff-Gustavson normal form and its convergence properties, J. Phys. A Math. Gen. 25 (1992) 5311-5327. 

R. T. Swimm and J. B. Delos, Semiclassical calculations of vibrational energy levels for nonseparable systems using the Birkhoff-Gustavson normal form, J. Chem. Phys. 71 1706 (1979). 

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