Read e-book online Differential Galois Theory and Non-Integrability of PDF
By Juan J. Morales Ruiz
This ebook is dedicated to the relation among various options of integrability: the whole integrability of advanced analytical Hamiltonian platforms and the integrability of advanced analytical linear differential equations. For linear differential equations, integrability is made detailed in the framework of differential Galois idea. the relationship of those integrability notions is given by means of the variational equation (i.e. linearized equation) alongside a specific necessary curve of the Hamiltonian approach. The underlying heuristic notion, which stimulated the most effects offered during this monograph, is important for the integrability of a Hamiltonian process is the integrability of the variational equation alongside any of its specific vital curves. this concept ended in the algebraic non-integrability standards for Hamiltonian structures. those standards could be regarded as generalizations of classical non-integrability effects through Poincaré and Lyapunov, in addition to more moderen effects by way of Ziglin and Yoshida. therefore, via the differential Galois idea it isn't basically attainable to appreciate some of these methods in a unified method but additionally to enhance them. a number of very important purposes also are integrated: homogeneous potentials, Bianchi IX cosmological version, three-body challenge, Hénon-Heiles approach, etc.
The publication relies at the unique joint learn of the writer with J.M. Peris, J.P. Ramis and C. Simó, yet an attempt was once made to give those achievements of their logical order instead of their ancient one. the mandatory heritage on differential Galois concept and Hamiltonian structures is integrated, and a number of other new difficulties and conjectures which open new strains of study are proposed.
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The booklet is a wonderful creation to non-integrability equipment in Hamiltonian mechanics and brings the reader to the leading edge of analysis within the region. The inclusion of a big variety of worked-out examples, a lot of broad utilized curiosity, is commendable. there are various historic references, and an in depth bibliography.
For readers already ready within the prerequisite topics [differential Galois thought and Hamiltonian dynamical systems], the writer has supplied a logically available account of a notable interplay among differential algebra and dynamics.
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Extra resources for Differential Galois Theory and Non-Integrability of Hamiltonian Systems
Consequently the kΓ -bilinear map Ω; E ⊗ E → kΓ is non-degenerate. 3. Meromorphic connections 23 For many applications, we can identify the symplectic bundle V with the symplectic vector space E over the ﬁeld kΓ . In this situation all the purely algebraic results on symplectic vector spaces over the numerical ﬁelds R or C remain also true . , canonical frames given by global meromorphic sections, and, with respect to a symplectic base, Ω is represented by the canonical form J= 0 −I I 0 . Furthermore, changes of symplectic bases are given by elements of the symplectic group Sp(n, kΓ ) ⊂ GL(2n, kΓ ).
We denote the ground numerical field by k. So, k = R in the real case and k = C in the complex case. This chapter can not be considered as an introduction to Hamiltonian systems. As in Chapter 2, we have, in general, presented only the required J. J. 1007/978-3-0348-0723-4_3, © Springer Basel 1999 43 44 Chapter 3. Hamiltonian Systems deﬁnitions and results without proofs. 4 about the abelian structure of some subalgebras of the Poisson algebra of rational functions. P. Ramis  and they are essential preliminaries for proof of the fundamental theorems of Chapter 4.
We will say that it is holomorphic at a point p ∈ Γ if, for every germ at p of the holomorphic vector ﬁeld X, the space of germs at p of holomorphic sections of the ﬁbre bundle V is invariant by the covariant derivative ∇X . Later we will consider connections that are meromorphic on Γ and holomorphic on Γ. They can have poles on the singular set S. If we want to compute in local coordinates in a neighborhood of a singular point s ∈ S, then we choose a holomorphic coordinate t at s (vanishing at s) d and we write our given vector ﬁeld X = f (t) dt , where f ∈ ks (in general we d cannot write X as dt , because the ﬁeld X may vanish or admit a pole at the point s, as we shall see later in the applications).
Differential Galois Theory and Non-Integrability of Hamiltonian Systems by Juan J. Morales Ruiz