nach oben

1994 | Buch

Kapitel lesen Erstes Kapitel lesen

Dynamical Systems VII

Integrable Systems Nonholonomic Dynamical Systems

herausgegeben von: V. I. Arnol’d, S. P. Novikov

Verlag: Springer Berlin Heidelberg

Buchreihe : Encyclopaedia of Mathematical Sciences

Enthalten in: Professional Book Archive

Einloggen, um Zugang zu erhalten

Inhaltsverzeichnis

Frontmatter

Nonholonomic Dynamical Systems, Geometry of Distributions and Variational Problems

Frontmatter

Introduction

Abstract

A nonholonomic manifold is a smooth manifold equipped with a smooth distribution. This distribution is in general nonintegrable. The term ‘holonomic’ is due to Hertz and means ‘universal’, ‘integral’, ‘integrable’ (literally, ὁλoς -entire, voμoς - law). ‘Nonholonomic’ is therefore a synonym of ‘nonintegrable’.

V. I. Arnol’d, S. P. Novikov

Chapter 1. Geometry of Distributions

Abstract

In the sequel without further notice all objects, such as manifolds, functions, mappings, distributions, vector fields, forms, etc., are supposed to be infinitely differentiable.

V. I. Arnol’d, S. P. Novikov

Chapter 2. Basic Theory of Nonholonomic Riemannian Manifolds

Abstract

In this section we consider the nonholonomic variational problem on the minimum of length on a Riemannian manifold with constraints given by a nonholonomic distribution. The solutions to this problem, the nonholonomic geodesics, satisfy the Euler-Lagrange equations of a conditional problem. They generate a nonholonomic geodesic flow defined on the mixed bundle which is the direct sum of the distribution and its annihilator in the cotangent bundle (see Section 1.3). This flow allows to extend to the nonholonomic case the notion of exponential mapping.

V. I. Arnol’d, S. P. Novikov

Chapter 3. Nonholonomic Variational Problems on Three-Dimensional Lie Groups

Abstract

In this chapter we consider the simplest nonholonomic variational problems. We study three-dimensional nonholonomic Lie groups, i.e. groups with a left-invariant nonholonomic distribution. Our main subject is the study of the nonholonomic geodesic flow (NG-flow), more precisely, of the nonholonomic sphere, of the wave front (Section 1), and of the general dynamical properties of the flow (Section 2). The mixed bundle for Lie groups is the direct product G × (V ⊕ V⊥). In Section 1.1 we show that the NG-flow on the mixed bundle is the semidirect product with base V ⊕ V⊥ and fiber G. In Section 1.2 we describe left-invariant metric tensors on Lie algebras; in Section 1.3 the normal forms for the equations of nonholonomic geodesics are obtained. In Section 1.4 we study the reduced flow on V ⊕ V ^⊥. In the subsequent Sections (1.5–1.7) we describe local properties of the flow on the fiber; in Section 1.5 we describe the ε-wave front of the NG-flow and the ε-sphere of the nonholonomic metrics which appear to be manifolds with singularities, the same for all three-dimensional nonholonomic Lie groups. In Section 1.6 we describe their topology and in Section 1.7 their metric structure.

V. I. Arnol’d, S. P. Novikov

Integrable Systems II

Frontmatter

Introduction

Abstract

The three chapters that follow are conceived as independent surveys dealing with various group-theoretical constructions of finite-dimensional integrable systems. The major part of Chapter 1 is devoted to the so-called Calogero-Sutherland systems, which historically are among the first examples of this kind. A somewhat wider range of applications is provided by the Kostant-Adler scheme and its generalization known as the r-matrix construction; these are discussed in Chapter 2 with an emphasis on concrete examples. A natural class of Lie algebras where the r-matrix construction gives interesting results includes semi-simple Lie algebras and loop algebras (or affine Lie algebras); the latter appear already in Chapter 1 in connection with periodic Toda lattices and are exploited more profoundly in Chapter 2. The r-matrix approach applied to loop algebras gives a natural passage to algebraic-geometric methods; explanation of these links is another major theme of Chapter 2. (We also recommend it to the reader to consult the survey “Integrable systems. 1” by B. Dubrovin, I. Krichever and S. Novikov, EMS vol. 4, Springer-Verlag 1990.) Finally, Chapter 3 is concerned entirely with the quantization problem for a particular, though very interesting, family of integrable systems, the nonperiodic Toda lattices.

V. I. Arnol’d, S. P. Novikov

Chapter 1. Integrable Systems and Finite-Dimensional Lie Algebras

Abstract

In this survey we consider integrable systems whose construction makes use of root systems of simple (usually finite-dimensional) Lie algebras.

M. A. Olshanetsky, A. M. Perelomov

Chapter 2. Group-Theoretical Methods in the Theory of Finite-Dimensional Integrable Systems

Abstract

The present survey is devoted to a general group-theoretic scheme which allows to construct integrable Hamiltonian systems and their solutions in a systematic way. This scheme originates from the works of Kostant [1979a] and Adler [1979] where some special but very instructive examples were studied. Some years later a link was established between this scheme and the so-called classical R-matrix method (Faddeev [1984], Semenov-Tian-Shansky [1983]). One of the advantages of this approach is that it unveils the intimate relationship between the Hamiltonian structure of an integrable system and the specific Riemann problem (or, more generally, factorization problem) that is used to find its solutions. This shows, in particular, that the Hamiltonian structure is completely determined by the Riemann problem. The simplest system which may be studied in this way is the open Toda lattice already described in Chapter 1 by Olshanetsky and Perelomov. (The Toda lattices will be considered here again in a more general framework.) However, the most interesting examples are related to infinite-dimensional Lie algebras. In fact, it can be shown that the solutions of Hamiltonian systems associated with finite-dimensional Lie algebras have a too simple time dependence (roughly speaking, like trigonometric polinomials). By contrast, genuine mechanical problems often lead to more sophisticated (e.g. elliptic or abelian) functions.

A. G. Reyman, M. A. Semenov-Tian-Shansky

Chapter 3. Quantization of Open Toda Lattices

Abstract

In the previous chapter we described a geometric realization of open Toda lattices. Already in his first paper on the subject Kostant [1979] indicated an extension of geometrical scheme to quantum Toda lattices. About the same time (1978–1981), a powerful and sophisticated technique of the Quantum Inverse Scattering Method was created for the study of quantum integrable systems (see Faddeev [1980, 1984]). As was soon realized, it goes beyond the ordinary theory of Lie groups and Lie algebras and introduces the new notion of quantum groups (Drinfel’d [1987]). From this general point of view the ‘quantum Kostant-Adler scheme’ is a certain limiting case which corresponds to linearization of the basic commutation relations. [It should be recalled that the basic commutation relations of the Quantum Inverse Scattering Method are quadratic; they may be regarded as the quantization of quadratic Poisson bracket relations considered in Section 12 of the previous chapter.]

M. A. Semenov-Tian-Shansky

Geometric and Algebraic Mechanisms of the Integrability of Hamiltonian Systems on Homogeneous Spaces and Lie Algebras

Frontmatter

Chapter 1. Geometry and Topology of Hamiltonian Systems

Abstract

A pair (M, ω) consisting of a 2n-dimensional manifold M together with a closed 2-form ω is called a symplectic manifold if the form ω is nondegenerate, i.e. if ω ⁿ = ω ∧ · ... · ω ≢ 0.

V. I. Arnol’d, S. P. Novikov

Chapter 2. The Algebra of Hamiltonian Systems

Abstract

It is well known that the projective space ℂP ⁿ carries a canonical symplectic structure (see, for instance, Arnol’d [1974] or Arnol’d and Givental’ [1985]). We consider a representation ρ: G →GL(V) of a Lie group G on a finite-dimensional complex vector space V. There is an induced action of G on the projective space P(V) associated with V. We denote by G(z) the orbit of this action passing through a point z ϵ P(V). It is now natural to ask when the symplectic structure of P(V) induces a symplectic structure on the orbits of G in P(V). Let us assume that G is compact, let T be a maximal torus in G, let g and t be the Lie algebras of G and T, respectively, and let L ^ℂ denote the complexification of a vector space L. We then have the root space decomposition \( {g^C} = {t^C} + \sum\limits_{\alpha \ne 0} {\mathbb{C}{E_\alpha }{H_\alpha }} = \left[ {{E_\alpha }{E_{ - \alpha }}} \right] \) (see, for instance, Helgason [1962], Jacobson [1962] or the Appendix). Let n: V\0 → P(V) be the natural projection which maps a vector υ ∊ V\0 into the straight line passing through υ and the origin.

V. I. Arnol’d, S. P. Novikov

Backmatter

Titel: Dynamical Systems VII
herausgegeben von: V. I. Arnol’d
S. P. Novikov
Verlag: Springer Berlin Heidelberg
Electronic ISBN: 978-3-662-06796-3
Print ISBN: 978-3-642-05738-0
DOI: https://doi.org/10.1007/978-3-662-06796-3

Springer Professional

Inhaltsverzeichnis

Frontmatter

Nonholonomic Dynamical Systems, Geometry of Distributions and Variational Problems

Frontmatter

Introduction

Chapter 1. Geometry of Distributions

Chapter 2. Basic Theory of Nonholonomic Riemannian Manifolds

Chapter 3. Nonholonomic Variational Problems on Three-Dimensional Lie Groups

Integrable Systems II

Frontmatter

Introduction

Chapter 1. Integrable Systems and Finite-Dimensional Lie Algebras

Chapter 2. Group-Theoretical Methods in the Theory of Finite-Dimensional Integrable Systems

Chapter 3. Quantization of Open Toda Lattices

Geometric and Algebraic Mechanisms of the Integrability of Hamiltonian Systems on Homogeneous Spaces and Lie Algebras

Frontmatter

Chapter 1. Geometry and Topology of Hamiltonian Systems

Chapter 2. The Algebra of Hamiltonian Systems

Backmatter