Physics
Integrable Systems
Integrable systems are physical systems that can be solved exactly using mathematical methods. They are characterized by having a sufficient number of conserved quantities, which allows for the system's behavior to be predicted with precision. Integrable systems have applications in various fields, including classical mechanics, quantum mechanics, and statistical mechanics.
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7 Key excerpts on "Integrable Systems"
eBook - PDF- T. N. Bailey, R. J. Baston(Authors)
- 1990(Publication Date)
- Cambridge University Press(Publisher)
Partly because the subject involves a great deal of very beautiful mathematics. And partly because integrable equations are relevant to the real world, in describing real phenomena such as solitons, or in serving as a first approximation to a more accurate model. The twistor description has turned out to be particularly appropriate for many (if not all) classical Integrable Systems. (It has not, as yet, had much impact in the area of integrable quantum systems.) This review will attempt to show how the plethora of known Integrable Systems are related to one another, and how they fit into the twistor framework. 2 What is Integrability? Let us begin by considering ordinary differential equations. In classical me- chanics, the standard 'Liouville' definition of integrability is that there should exist a sufficient number of constants of motion, enabling one to reduce the equations of motion to quadratures (see, for example, [2]). If the phase space is 2m-dimensional, then one wants m independent integrals of the motion, in involution with one another. It is crucial that these should exist globally, i.e. that they be smooth, single-valued functions on phase space. And one can add various other technical requirements. One particularly strong version is that of 'algebraic complete integrability' (see [23]). This involves complexifying the system, so that the phase-space variables become holomorphic functions of 'complex time'. And integrability then means that the time-evolution of the system consists of linear flow on m-dimensional abelian varieties (complex algebraic tori) in complexified phase space (these tori are the level sets of the m constants of motion). §2 What is Integrability? 247 There is a well-known way of generating systems of ordinary differential equations having some constants of motion. Let L be an n x n matrix of functions of t; and let P be another matrix, the elements of which are given by some local expressions in terms of the elements of L.
eBook - PDF- John H. Lowenstein(Author)
- 2012(Publication Date)
- Cambridge University Press(Publisher)
3 Integrable Systems A Hamiltonian system with 2n-dimensional phase space is called integrable if there exist n independent, mutually commuting, smooth functions F 1 , . . . , F n and a Hamiltonian H = H ( F 1 , . . . , F n ). Owing to the Poisson bracket relations [ F k , H ] = 0 = dF k dt , k = 1, . . . , n, the F k are integrals, i.e. constants of the motion along phase-space orbits. Without loss of generality, we may assume that one of these, F 1 , is H itself. Among all possible Hamiltonian systems, the integrable ones are exceedingly rare. Nonetheless they occupy a privileged place in classical dynamics. One reason is that essentially all systems that are explicitly soluble using the methods of tra- ditional mathematical physics [2] are integrable. Most of the examples which we have encountered in this book so far have been in this category. Moreover, typical attempts to understand non-integrable models involve using a form of perturbation theory in which the unperturbed system is an integrable one. For these reasons, we give considerable attention to the construction and analysis of Integrable Systems in this book. 3.1 The Liouville–Arnol’d theorem What is special about Integrable Systems? A remarkably detailed answer to this question is provided by the Liouville–Arnol’d theorem [6, 7], which states that a system with independent commuting integrals F k , k = 1, . . . , n satisfies the following: 1. The phase space is partitioned into level sets M f = {ξ : F k (ξ) = f k , k = 1, . . . , n}, each of which is invariant under the Hamiltonian flows generated by the n functions F k . This includes the time evolution generated by H = F 1 . For 56 3.2 Fast track for separable systems 57 simplicity, we assume that M f is connected; otherwise we consider separately each connected component of M f . 2. For given f = ( f 1 , . . . , f n ), M f is diffeomorphic to T m × R n−m , where T m is a torus of dimension m, m ∈ {0, .
eBook - PDF- Jean-marc Ginoux(Author)
- 2009(Publication Date)
- World Scientific(Publisher)
In this aim, many methods have been developed the two main of which consist in solving differential equations either by series expansions (New-ton, Leibniz, Euler, . . . ) or explicitly by “ quadratures” (closed-form), i.e. while using a combination of elementary functions divided into algebraic and transcendental categories. The former, also known as perturbation expansions, fails in the vicinity of singular points as pointed out by Henri Poincar´ e in his memoirs (Poincar´ e, 1886) while for the later Joseph Liouville has stated in a note (Liouville, 1839) that the solution of any differential equation can not be necessary expressed in terms of elementary functions . Then, the notion of integrability was introduced to distinguish Integrable Systems from non-integrable . 6.1 Integrability conditions, integrating factor, multiplier In the first part of this book, Sec. 1.2, it has been recalled that, accord-ing to D’Alembert transformation, a n th order differential equation can be transformed into a system of n simultaneous first-order equations, i.e. into a dynamical system . It will be established in this section that a total differ-85 86 Differential Geometry Applied to Dynamical Systems ential equation , i.e. a differential 1-form provides another way of obtaining a dynamical system . Moreover, integrability conditions of such equation will be used to state about the integrability of the corresponding dynamical system . 6.1.1 Two-dimensional dynamical systems Let’s consider a first order and first degree total differential equation in-volving two variables, x and y which represents a differential 1-form : dφ = P ( x, y ) dx + Q ( x, y ) dy = 0 (6.1) where P and Q are supposed to be C ∞ continuous functions in E with values in R , checking the assumptions of the Cauchy-Lipschitz theorem. General integral of this differential equation reads: φ ( X ) = φ ( x, y ) = C and defines a family of plane curves depending on an arbitrary constant C .
eBook - PDF- W. F. Ames(Author)
- 1991(Publication Date)
- Academic Press(Publisher)
Hamiltonian Structure and Integrability Benno Fuchssteiner University of Paderborn D 4790 Paderhorn Germany 1 Iiitroductioii Whenever a quantity, or a set of quantities, evolves with time then we call this a dynamical system. The evolution of tlie universe certainly is a. dyna.mical system, however a compli- cated one. The la.ws of evolution which govern such a system are called the dynamical laws. To describe dyiianucal systcnis we itsiially make suita.ble approximations i n the hope of finding valid descriptions of their clia.racteristic quantities. But even after such approx- imations we mostly cannot write down explicitly liow these quantities depend on time, usually sucli a dependeiice is niucli to9con~plicat~ed to be cotuputed explicitly. Therefore we commonly write down dynamical systcnis i n their infinitesimal form. Considering a dynainical system i n its infinitesimal form has many advantages. The principal one is that sucli an infiiiitcsiinal description is possible even in those caws where a global description is not feasihlc at. all. Tecltnically speaking, an infinitesimal description leads to a diffcrential equa.t,ion,which in many cases h a s nonlinear terms due to the interac- tion between different quantit,ies. To find sucli a. differential equation we only have to know a suitable set of dynamical laws. Ilowever. solving such a nonlinear differential equation for arhitrary sta.rting points (initial coritlitions) is often a ho~~eless endeavor. Fortunately, the infinitesimal tlescriptioii sometimes gives an insight into the essential structures for the dyna.mics of t.hc syst,eni, or a t least into those parts of the dynamics which ca.n he described locally. Speaking from a.n abstract. viewpoint the niain objects of our interest are equations of the form tit = 1i(u) (1.1) where I<(%) is a vector field oii sonie inanifold Af a,nd where 11 denotes the general point on this manifold.
eBook - PDF- Michael Atiyah, Daniel Iagolnitzer(Authors)
- 1997(Publication Date)
- WSPC(Publisher)
A lot of new things indeed appeared during the eighties, as I will next explain. Thus I started producing new work in topology in the past decade only. Before that I sold my knowledge about topology to physicists. In fact, dynamical systems was more my own way of solving models. The new direction in my work started with the discovery of soliton theory, which was done in the late sixties. It was an extremely interesting finding, which led, among other things, to modern Integrable Systems, conformal field theory, and quantum group theory (as a late by-product). Integrable Models in Classical Mathematics and Mechanics Everyone knows the role of the famous two-body problem, solved by Newton, in the development of the mathematical methods of physics. For a long period after that, people used the method of the exact analytic solution for some differential equations as a principal tool in mathematical physics. They simplified their problem if it was (or looked) too difficult, and after that tried to find the exact solution. 210 Fields Medallists' Lectures A lot of work has been done in the process of searching for special integrable cases of famous problems, like the motion of the top, for example. All mathematical methods — like power and trigonometric series, Fourier-Laplace (and other) integral transformations, complex analysis and symmetry arguments — were discovered and developed for that in the nineteenth century. These methods led sometimes to remarkable negative results, i.e., to proofs that certain models are not solvable in principle. Some strange integrable cases which do not admit any obvious symmetry were discovered in the nineteenth century: the integrability of geodesies on 2-dimensional ellipsoids in Euclidean 3-space (Jacobi), the motion of the top with special param-eters for constant gravity (Kovalevskaya), and some others. Riemann surfaces and ^-functions of genus 2 played the leading role in their integrability.
- Giovanni Giachetta, Luigi Mangiarotti, Gennadi A Sardanashvily(Authors)
- 2010(Publication Date)
- World Scientific(Publisher)
7.1.1 Partially Integrable Systems on a Poisson manifold Completely integrable and superIntegrable Systems are considered with re-spect to a symplectic structure on a manifold which holds fixed from the beginning. A partially integrable system admits different compatible Pois-son structures (see Theorem 7.1.2 below). Treating partially integrable sys-tems, we therefore are based on a wider notion of the dynamical algebra [62; 65]. Let we have m mutually commutative vector fields { ϑ λ } on a connected smooth real manifold Z which are independent almost everywhere on Z , i.e., the set of points, where the multivector field m ∧ ϑ λ vanishes, is nowhere dense. We denote by S ⊂ C ∞ ( Z ) the R -subring of smooth real functions f on Z whose derivations ϑ λ c df vanish for all ϑ λ . Let A be an m -dimensional Lie S -algebra generated by the vector fields { ϑ λ } . One can think of one of its elements as being an autonomous first order dynamic equation on Z and of the other as being its integrals of motion in accordance with Definition 1.10.1. By virtue of this definition, elements of S also are regarded as integrals of motion. Therefore, we agree to call A a dynamical algebra . Given a commutative dynamical algebra A on a manifold Z , let G be the group of local diffeomorphisms of Z generated by the flows of these vector fields. The orbits of G are maximal invariant submanifolds of A (we follow the terminology of [153]). Tangent spaces to these submanifolds form a (non-regular) distribution V ⊂ TZ whose maximal integral manifolds co-incide with orbits of G . Let z ∈ Z be a regular point of the distribution V , i.e., m ∧ ϑ λ ( z ) 6 = 0. Since the group G preserves m ∧ ϑ λ , a maximal integral manifold M of V through z also is regular (i.e., its points are regular). Fur-thermore, there exists an open neighborhood U of M such that, restricted to U , the distribution V is an m -dimensional regular distribution on U .
eBook - PDF- Ahmed Lesfari(Author)
- 2022(Publication Date)
- Wiley-ISTE(Publisher)
3 Integrable Systems The aim of this chapter is to study the Arnold–Liouville theorem and its connection with completely Integrable Systems. Many Integrable Systems are studied in detail, including: the problem of the rotation of a rigid body about a fixed point – the Euler problem of a rigid body, the Lagrange top, the Kowalewski top and other special cases, such as the Hesse–Appel’rot top, Goryachev–Chaplygin top and Bobylev–Steklov top; the problem of motion of a solid in an ideal fluid – Clebsch’s case and Lyapunov– Steklov’s case; the Yang–Mills field with gauge group SU (2). Some of these problems will be studied in detail in other chapters, using other methods. 3.1. Hamiltonian systems and Arnold–Liouville theorem DEFINITION 3.1.– A Hamiltonian system is a triple (M, ω, H), where (M, ω) is a 2n-dimensional symplectic manifold (the phase space) and H ∈ C ∞ (M ) is a smooth function (Hamiltonian). Recall section 1.5, where we showed that we have a complete characterization of Hamiltonian vector field ˙ x(t) = X H (x(t)) = J ∂H ∂x , x ∈ M, [3.1] where H : M −→ R is the Hamiltonian, J = J (x) is a skew-symmetric matrix, possibly depending on x ∈ M , and for which the corresponding Poisson bracket {H, F } = ∑ i,j J ij ∂H ∂x i ∂F ∂x j satisfies the Jacobi identity: {{H, F }, G}+ {{F, G}, H} + {{G, H}, F } = 0. DEFINITION 3.2.– A Hamiltonian system [3.1] is called Liouville integrable or completely integrable if it possesses n smooth functions (first integrals or constants of motion), H 1 = H, H 2 , ..., H n defined on M , such that (i) H 1 , H 2 , ..., H n are in Integrable Systems, First Edition. Ahmed Lesfari. © ISTE Ltd 2022. Published by ISTE Ltd and John Wiley & Sons, Inc. 68 Integrable Systems involution, that this {H i , H j } = 0, i, j = 1, ..., n, and (ii) H 1 , H 2 , ..., H n are functionally independent. In other words, the differentials dH 1 , ..., dH n are linearly independent on a dense open subset of M or dH 1 ∧ ...
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