By Juan J. Morales Ruiz

This booklet is dedicated to the relation among diversified suggestions of integrability: the total integrability of advanced analytical Hamiltonian platforms and the integrability of advanced analytical linear differential equations. For linear differential equations, integrability is made certain in the framework of differential Galois concept. the relationship of those integrability notions is given through the variational equation (i.e. linearized equation) alongside a selected fundamental curve of the Hamiltonian procedure. The underlying heuristic inspiration, which prompted the most effects offered during this monograph, is valuable situation for the integrability of a Hamiltonian approach is the integrability of the variational equation alongside any of its specific quintessential curves. this concept resulted in the algebraic non-integrability standards for Hamiltonian platforms. those standards will be regarded as generalizations of classical non-integrability effects via Poincaré and Lyapunov, in addition to more moderen effects through Ziglin and Yoshida. therefore, through the differential Galois idea it isn't in simple terms attainable to appreciate a majority of these methods in a unified means but in addition to enhance them. numerous vital functions also are integrated: homogeneous potentials, Bianchi IX cosmological version, three-body challenge, Hénon-Heiles method, etc.

The e-book relies at the unique joint study of the writer with J.M. Peris, J.P. Ramis and C. Simó, yet an attempt used to be made to give those achievements of their logical order instead of their old one. the mandatory heritage on differential Galois conception and Hamiltonian platforms is incorporated, and several other new difficulties and conjectures which open new traces of analysis are proposed.

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The e-book is a wonderful creation to non-integrability equipment in Hamiltonian mechanics and brings the reader to the leading edge of analysis within the quarter. The inclusion of a giant variety of worked-out examples, lots of vast utilized curiosity, is commendable. there are lots of historic references, and an in depth bibliography.

(Mathematical Reviews)

For readers already ready within the prerequisite topics [differential Galois thought and Hamiltonian dynamical systems], the writer has supplied a logically obtainable account of a outstanding interplay among differential algebra and dynamics.

(Zentralblatt MATH)

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**Extra info for Differential Galois Theory and Non-Integrability of Hamiltonian Systems**

**Example text**

We get the usual musical map by restriction to M . This map is an isomorphism over M . Writing the application in coordinates in a neighborhood of a point at inﬁnity we see that admits an inverse. This inverse is holomorphic over M but can have poles over M∞ . We ﬁnish this section with two remarks. So far we have only considered autonomous Hamiltonian systems (Hamiltonian vector ﬁelds independent of time). 1 of [3]). From an intrinsic point of view this is done 48 Chapter 3. Hamiltonian Systems in the context of contact geometry, instead of symplectic geometry, which, in coordinates, is given by the extended symplectic form n dyi ∧ dxi − dh ∧ dt, i=1 deﬁned in the 2n+1-dimensional extended phase space parametrized by (xi ,yi ,t) (i = 1, .

All the above constructions remain valid if we start with a local meromorphic connection on the vector space V over the ﬁeld C{t}[t−1 ] with the d suitable dictionary: dt instead X, etc. . 24 Chapter 2. 4 The Tannakian approach We present now the Galois theory from the intrinsic connection perspective [21, 29, 51, 69]. Let (V, ∇) be, as in the above section, a meromorphic connection over a ﬁbre bundle of rank m. Then, we consider the horizontal sections, Sol ∇ := Solp0 ∇ of this connection at a ﬁxed non-singular point p0 ∈ Γ (they correspond to solutions of the corresponding linear equation).

In particular, they belong to SL(2, C). The complex numbers µ, λ are called the Stokes multipliers. In particular they belong to SL(2, C). 6) dx (see [74, 16]). 6) is topologically generated by the exponential torus, the formal monodromy and the Stokes matrices (at x = 0). We note that among these generators the main source of non-integrability comes from the Stokes multipliers. For example, it is not diﬃcult to prove that the Zariski closure of the group (algebraically) generated by the two matrices 1 0 λ 1 , 1 0 µ 1 , where λ, µ are both diﬀerent from zero, is SL(2, C) [18].