Integrability, chaos, and complexity

Phase space flows divide into two basic classes: completely integrable and ‘nonintegrable’ (incompletely integrable). In the first case the motion is equivalent to a single time-translation for all finite times (chapter 3). In the second case singularities of the conservation laws prevent this degree of simplicity (section 13.2).

Nonintegrable systems can be divided further into two classes: those that are in principle predictable for all times, and those that are not. The first class includes deterministic dynamics where there is a generating partition. An example, the asymmetric tent map, was given in chapter 14. A chaotic dynamical system generates infinitely many different statistical distributions through the variation of classes of initial conditions. Only the differentiable distribution (corresponding to the invariant density) is generated by initial data that occur with measure one. The fragmented statistical distributions, including multifractal ones, occur for the measure zero set of initial conditions.

Whenever a dynamical system is computable (McCauley, 1993), which is the only case that can be used computationally and empirically, then the measure zero initial conditions are of dominant importance. There is, as yet, no known case in nature where the differentiable probability distribution describes the experiments or observations performed on a driven-dissipative chaotic dynamical system. The generating partition forms the support of every probability distribution that occurs mathematically, and can be discovered (when it exists) through backward iteration of the Poincaré map.

Overview

In this chapter we return to transformations among Cartesian frames of reference. In the first chapter we studied physics in inertial and linearly accelerated frames. Here, we formulate physics in rotating frames from both the Newtonian and Lagrangian standpoints.

Once one separates the translational motion of the center of mass then rigid body theory for a single solid body is the theory of rigid rotations in three dimensions. The different coordinate systems needed for describing rigid body motions are the different possible parameterizations of the rotation group. The same tools are used to describe transformations among rotating and nonrotating frames in particle mechanics, and so we begin with the necessary mathematics. We introduce rotating frames and the motions of bodies relative to those frames in this chapter and go on to rigid body motions in the next. Along the way, we will need to diagonalize matrices and so we include that topic here as well.

In discussing physics in rotating frames, we are prepared to proceed in one of two ways: by a direct term by term transformation of Newton's laws to rotating frames, or by a direct application of Lagrange's equations, which are covariant. We follow both paths in this chapter because it is best to understand physics from as many different angles as possible. Before using Lagrange's equations, we also derive a formulation of Newton's equations that is covariant with respect to transformations to and among rotating Cartesian frames.

In variance principles and integrability (or lack of it) are the organizing principles of this text. Chaos, fractals, and strange attractors occur in different nonintegrable Newtonian dynamical systems. We lead the reader systematically into modern nonlinear dynamics via standard examples from mechanics. Goldstein and Landau and Lifshitzpresume integrability implicitly without defining it. Arnol'd's inspiring and informative book on classical mechanics discusses some ideas of deterministic chaos at a level aimed at advanced readers rather than at beginners, is short on driven dissipative systems, and his treatment of Hamiltonian systems relies on Cartan's formalism of exterior differential forms, requiring a degree of mathematical preparation that is neither typical nor necessary for physics and engineering graduate students.

The old Lie-Jacobi idea of complete integrability (‘integrability’) is the reduction of the solution of a dynamical system to a complete set of independent integrations via a coordinate transformation (‘reduction to quadratures’). The related organizing principle, invariance and symmetry, is also expressed by using coordinate transformations. Coordinate transformations and geometry therefore determine the method of this text. For the mathematically inclined reader, the language of this text is not ‘coordinatefree’, but the method is effectively coordinate-free and can be classified as qualitative methods in classical mechanics combined with a Lie–Jacobi approach to integrability. We use coordinates explicitly in the physics tradition rather than straining to achieve the most abstract (and thereby also most unreadable and least useful) presentation possible.

Solvable vs integrable

In this chapter we will consider n coupled and generally nonlinear differential equations written in the form dxi/dt = Vi(x1,…,xn). Since Newton's formulation of mechanics via differential equations, the idea of what is meant by a solution has a short but very interesting history (see Ince's appendix (1956), and also Wintner (1941)). In the last century, the idea of solving a system of differential equations was generally the ‘reduction to quadratures’, meaning the solution of n differential equations by means of a complete set of n independent integrations (generally in the form of n – 1 conservation laws combined with a single final integration after n – 1 eliminating variables). Systems of differential equations that are discussed analytically in mechanics textbooks are almost exclusively restricted to those that can be solved by this method. Jacobi (German, 1804–1851)systematized the method, and it has become irreversibly mislabeled as ‘integrability’. Following Jacobi and also contemporary ideas of geometry, Lie (Norwegian, 1842–1899) studied first order systems of differential equations from the standpoint of invariance and showed that there is a universal geometric interpretation of all solutions that fall into Jacobi's ‘integrable’ category.

Galileo on falling bodies

In 1638 the Dutch publishers Elzivir† published a book by Galileo entitled Dialogues Concerning Two New Sciences. Since the Catholic Church had put Galileo under permanent house arrest and forbidden the publication of any book written by him, the work is introduced by a preface in which Galileo expresses surprise that a manuscript intended for a few private friends should have found its way into the hands of the printers. In spite of the difficult circumstances of its composition‡, the book sparkles with good humour. It takes the form of a dialogue between three friends: Salviati, who puts the point of view of Galileo's new physics, Simplicio, who puts the old point of view and Sagredo, who represents the intelligent layman. Here they discuss Aristotle's view that things fall at a speed proportional to their weight.

SALVIATI … I greatly doubt that Aristotle ever tested by experiment whether it be true that two stones, one weighing ten times as much as the other, if allowed to fall at the same instant, from a height of, say, 100 cubits, would so differ in speed that, when the heavier had reached the ground, the other would not have fallen more than 10 cubits.

SIMPLICIO His language would seem to indicate that he had tried the experiment, because he says: We see the heavier; now the word see shows that he had made the experiment.