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The Algebraic “Main Step” Map
Given a Lie algebra V with bracket f g, to define a dynamical system on it we introduce the space derV of derivations of V, derV def = n D: V ! V s.t. DfFgG = fDFgG + fFgDG; 8F;G 2 V o (2.1) Inner derivations are those for which D fDg. Derivations which are not inner are called outer. Once a derivation H is fixed, a dynamical system is given by f_ = Hf (2.2).
A perturbation of this dynamical system is H ! H + fV g In our perturbation theory, the original derivation H can be of any type, also outer, but the perturbation must be inner. Another tool that will be needed is a scale of norms (described in detail in A.1.1). A norm is a function V ! R+, with two properties of triangular inequality and linearity: kF + Gk kFk + kGk ; kFk = jjkFk ; 8F 2 V; 2 K
A scale of norms is a sequence of norms k krr2I, where I is an ordered set, such that kFkr1 kFkr2 ; 8r1; r2 2 I ; r1 r2 In the next proposition, we propose a formula, henceforth called “the main step formula”, to conjugate the perturbed derivation H + fV g to another one, where the perturbation V is split into a term preserving the same subalgebra of H, plus a new perturbation V, smaller than V . The conjugation is defined by a Lie series. By assuming that there exist a scale of Banach norms (see appendix A.1.1) defined on V, we give minimal hypothesis to ensure the convergence of the series. In figure 2.1 we provide a scheme of all the quantities involved in our formula.
A brief historical survey
The literature on KAM theory is immense; here we propose a modest review without any claim of completeness, focusing on those developments we met in developing our work. In 1954 Kolmogorov proposed a new way to overcome the convergence issues of classical perturbation theory [47]. The main novelty of this approach was to focus on a single torus, and to build a superconvergent algorithm which conjugates the perturbed Hamiltonian to a new one, preserving the prescribed torus. Convergence was assured by the “Diophantine condition” and by non-degeneracy of the Hamiltonian. Also, the Hamiltonian was required to be analytic. The proof was so concise that it was accused to be incomplete [3]. However, a new proof of the theorem, following the original guidelines of Kolmogorov, was provided by Benettin, Galgani, Giogilli and Strelcyn in 1984 [7].
In 1963 Arnol’d [3], [4] gave a new proof of the theorem, with a few technical differencies. Also, he dropped the non-degeneracy condition assumed by Kolmogorov, in place of another one, called the isoenergetic non-degeneracy. In 1962 also Moser provided a result analogous to Kolmogorov’s theorem in the framework of area-preserving twist mappings of an annulus ( [68], [71]), dropping the analiticity condition for finite differentiability (up to order 333). In the context of Hamiltonian mechanics the requirement for analiticity of the perturbation was dropped by Poschel [78]; later by Salamon dropped also the analiticity of the Hamiltonian [83]. The Diophantine condition was relaxed by Russman [81] and even further by Bruno [94].
One line of research in KAM theory has been the development of an inversion theorem for Frechet spaces; it began with the works of Moser [70], [69], introducing the theorem nowadays called the Nash-Moser theorem. On this topic, we mention also the works of Zehnder [95], [96], Hamilton [45], Herman [8] and more recently Fejoz [37]. For a nice introduction see Raymond [79].
Another line of research was the relation between KAM and Renormalization theory: this was studied by Greene [42] and his student MacKay [57] for twist maps of the annulus, and by Gallavotti (see for instance [38]) and Doveil and Escande [29], [30], [27] for Hamiltonians. For the relation between these lines of research and plasma physics, see also [28].
Among the developments of KAM theory we mention also: a lagrangian formulation of the theorem proposed by Zehnder and Salamon [84]; a version adapted to nearly integrable systems on Poisson manifolds (based on action angle coordinates, but admitting a degeneracy in the symplectic form) by Li and Yi [52]; De La Llave, Gonzales, Jorba and Villanueva who proposed a KAM algorithm for Hamiltonian systems without action-angle variables [20]; De La Llave and Alishah for a KAM theorem for presymplectic systems [2]; Bounemura and Fischler, who recently proposed a new proof where diophantine estimates for small divisors are replaced by continued fractions approximations [9]. Finally we mention the works by Vittot [24], [92], [93]. He proposed an algebraic
perturbation theory, which required an operator G such that fHg2G = fHg; the reader can recognize a particular case of theorem 1, in the case of inner derivations (see remark 2), and having chosen B = kerfHg. However, instead of proposing an iteration mechanism à la Kolmogorov, the author proposes a formula to perform the perturbation transform in one step: given a perturbed system fH + V g, define W 2 V by V = F(W); F(W) = efGWgRW + 1 efGWg fGWg NW.
A scale of Banach norms for Vsymm
Functions in Vsymm admit the Fourier representation F(X; ; t) = X l;m2Z Fl;m(X) eilt+im (3.15). In analogy with classical mechanics, we build a complex extension of the domain of X. The procedure for the complexification of a Lie algebra is described in section A.2, however, we don’t need it as it’s evident that the complexification of R is C. For any X 2 (1; 1), we consider a ball Br(x) C of radius4 r 2 R+ centered at X. The radius r has to be sufficiently small so that jX rj < 1. Then we define the set Ar def = [ X2(1;1) Br(X).
Table of contents :
1 Introduction
2 Perturbation Theory on a Lie Algebra
2.1 Introduction
2.2 The Algebraic “Main Step” Map
2.2.1 A hint of the Iteration Procedure
2.3 A brief historical survey
3 The Throbbing Top
3.1 About the Rigid Body (or Top)
3.1.1 Lie-Poisson structure for the Rigid Body
3.1.2 The Throbbing Top
3.2 The symmetric case
3.2.1 A scale of Banach norms for Vsymm
3.2.2 Application of Proposition 1
3.2.3 A KAM theorem for the Symmetric Throbbing Top
3.2.4 Numerical experiments
3.2.5 About the Diophantine condition
3.3 The case of the Non-Symmetric Throbbing Top
3.3.1 A new scale of norms
3.4 Conclusion
4 The Charged Particle
4.1 The dynamical system
4.2 The Eulerian reduction
4.2.1 The Guiding Particle Solution
4.2.2 The Poisson structure after the Eulerian Reduction
4.2.3 Example: a constant uniform magnetic field
4.2.4 Another example: the toroidal field
4.2.5 Euler angles decomposition
4.3 Non-perturbative Guiding Centre Theory
4.3.1 The Guiding Centre Transform
4.3.2 From Lagrangian to Hamiltonian Formulation
4.3.3 Conservation of the Lagrangian
4.3.4 A hint for future developments
4.4 Conclusions
5 Conclusions and Future Perspectives
A Some mathematical tools
A.1 About Lie Algebras
A.1.1 Metric structure on the algebra
A.1.2 Non-autonomous Hamiltonian Systems
A.1.3 The Lie-Poisson theorem
A.2 Complexification of a Lie Algebra
A.3 About Rn
A.3.1 Vector fields
A.3.2 Coordinate systems on Rn
A.3.3 A special case: n = 3
B A WKB ansatz for equation (3.20)
C The Lagrangian description of dynamics
C.1 Relation with Hamiltonian formulation
C.2 Example: Relativistic Mechanics
C.3 The Poincaré-Cartan 1-form
