6,1. Hamiltonian systems

If dynamical systems can be shown to possess a Hamiltonian structure their analysis is greatlv facilitated [13]. In 1968 Misner [41] pointed out that the R oo Einstein equation (see eq. (3.6)), can be used to construct a Lagrangian whose variation then gives the remaining Rab equations. This develop­ment permitted a Hamiltonian formulation of the Einstein equations which was particularly useful in analysing the spatially homogeneous universes because they possess only a finite number of degrees of freedom. The cosmological problem of solving the field equations to determine the space-time metric components was seen to be equivalent to tracing the motion of a particle, or "universe point", within a potential well whose walls are determined by the form of the spatial three-curvature of the universe.

while the convergence of the geometric mean cycle length to the Kninchin constant * = 2.68... is given by an extension of (4.42) as

 

This representation was most conveniently couched in the Hamiltonian formalism developed earlier by Arcowitt, Deser and Misner [63]. They provided a series of reductions culminating in a represen­tation of the Einstein gravitational action in canonical form (for detailed descriptions of this procedure and the development of Hamiltonian cosmology see refs. [64]).

 

The scale factor 0(t) and shapes p± now parameterize the behaviour of the cosmological model. After further analysis [36,64] the form of the cosmological Hanrmtonian is found to be.

 

where p± are the momenta conjugate to and I2(t) should now be thought of as the time coordinate. Accordingly, Hamilton's equations are,

t In our case fi is diagonal but it will not be in general.

If we express the three-metric of eq. (3.7) in a form which separates the size and shape evolutions we have for the spatially homogeneous Bianchi universes,

 

 

where the 3x3 shape matrix is diagonal! and traceless. The scalar function /2(f) dec :ribes the volumetric behaviour and /2- r of section 4. The two independent shape parameters fin ,<nd are conventionally transformed to more convenient linear combinations, and /3_ where

 

The potential V(j8) is a positive definite combination of exponential factions which is determined by the appropriate Bianchi group structure constants of the homogeneous model in question. In the simplest Kasner (Bianchi I) Universe of section 3.1 the potential term (V-1) vanishes and then (6.4) describes the motion of a free particle. The one free Kasner index, or the parameter «, fixes the motion, with dfi+ldf} = (u2 + u - 0.5)/(m2+ u +1) and d/8_/drt = V3(u + 0.5)/(w2+ u +1). This tells us that when the universe point is well inside the potential walls, whatever their shape, the evolution will generally resemble that of the simple Kasner Universe. The potentials for the remaining homogeneous models are more complicated and are illustrated in fig. 4.

The most interesting cases, where the equipotentials are confining, arise for Bianchi types VIII and IX. For the diagonal models the exact form of their potential is where, iti the last term, the sign is taken (+) for type VIII and (-) for type IX.

Five points suffice to understand the coarse-grained behaviour of these systems:

All the potentials have exponentially steep walls with equipotentials forming equilateral triangles in the (0+, /8-) plane. However, the corners of the triangular contours are not closed, but rather have thin channels leading off to infinity (see fig. 4).

The potentials in fig. 4 and (6.6) are time-dependent. The walls expand as the Universe collapses to zero volume at t = 0. Encounters with walls become increasingly rare as /->0.

The reflections from the potential walls are not specular. The angle of incidence, 0U is related to the angle of reflection 0R by [64]

 

 

Within the Mixmaster (type IX) potential the universe point first undergoes a sequence of small oscillations against two potential walls until it enters into the corner channel formed by those two walls. Its motion is then reversed by the reflection (6.7) and it leaves the corner only to follow the same type of evolution against two different walls. The sequence of events continues ad infinitum in fl, or r, time. The sequence of small oscillations between pairs of potential walls corresponds to a cycle of evolution (displayed in fig. 3) during which two orthogonal axes oscillate whilst the third falls monotonically. The exchange to a new cycle of behaviour (x~* T(x)) corresponds to the departure from the corner channel. The non-zero metric entropy of the Mixmaster Universe is associated with the evolution out of the corner channels not the sequence of small oscillations between two walls.

The construction of a Poincare return map for Mixmaster was based upon its recurrent Kasner behaviour. This possibility is clearly illustrated in the Hamiltonian formulation where each period of evolution not involving collisions with walls is similar to the free particle motion of the Kasner model.

If we examine the collection of potentials Vifi) in fig. 4 we can determine the range of chaotic behaviour in the vacuum cosmological models they represent: Chaotic behaviour requires that neigh­bouring trajectories of universe points diverge as they are followed both forwards and backwards in time. This cannot occur in models where the potential is open in at least one direction (I—VII). The universe point will either approach its neighbours when evolved in the +D(t) or -0{t) directions. Chaotic behaviour is only possible in the VIII and IX models. The closed potentials allow recurrence to occur and the motion explores all of the phase space available to it. constrained or.lv bv the conservation laws of energy and momentum. The shape of the VIII and IX potentials in the corner channels supplies the instability necessary for chaotic behaviour. Of the homogeneous vacuum uni­verses, only Bianchi VIII and IX models can have non-zero metric entropies associated with their dynamics and in fact the metric entropies of these two models are equal.

There exists a clear connection between the Hamiltonian description of the VIII and IX models and the so-called 'Billiards' problem of ergodic theory which is concerned with the motion of particles inside a bounded region of R" (n 2), whose boundary is piece-wise smooth [65]. A dynamical system of

billiard type is generated by the uniform motion of a point «t constant velocity with elastic, specular reflections from both the walls and randomly placed obstacles within the potential boundaries. A

number of very general theorems is known for such systems and their elucidation is also linked to the properties of continued fraction expansions of functions. The detailed nature of these interconnections have not been worked out in detail.

A more straightforward problem is the connection between the Mixmaster Universe and another much-studied dynamical system: the geodesic flow on a negatively curved space [66]. It can be shown that the Mixmaster Hamiltonian system is asymptotically (on approach to the singularity with the number of oscillations becoming infinitely large, /}-+») equivalent to a geodesic flow on e space of negative curvature. This allows a number of the stochastic properties derived for the Mixmaster in section 4 to be arrived at in an independent manner.

 

where the new momentum component is just

 

The Misner transformation |69] introduces a set of hyperbolic coordinates (t, 6, <f>) in which the potential is time-independent after the substitutions

 

At large /3 the potential walls (6.6) have the form V(/3)~ as -». This wall becomes

 

The study of geodesic flows began with PoincarS's work [66]. Subsequently, Hadamaru [67] proved that the geodesic flow on a negatively curved Riemannian space is ergodic, Hedlund [68] showed that is is strongly mixing and recently Ornstein and Weiss [69] have proved the Bernoulli property. It is clear why such stochastic properties arise: the separation, A, of neighbouring geocesics moving on a space of curvature K(t) obeys the Jacobi equation,

 

 

Every solution of this equation diverges to infinity as f-»±°o when K<0: Neighbouring geodesies diverge exponentially and exhibit sensitive dependence upon initial conditions. This is clearly not the case when K 3s 0.

To demonstrate the equivalence of the Mixmaster Universe to geodesic flows of this sort we need to remove the time-dependence in the curvature potential V(/L). The existence of such a transformation has been demonstrated by Misner [69]. Defining a new time coordinate A by dA = d fl/H a new system of Hamilton's equations are obtained from (6.5) with a new Hamiltonian, //, given byBumm,  and in the asymptotic limit on approach to the singularity R(6, <f>) becomes

Hit asymptotic behaviour is controlled by the quantity in (. . .) and so one stationary asymptotic wall of the potential is defined by

 

 

\nd the other two walls are obtained from this by the transformations <f> $ ± 2w/ >. The Hamiltonian is tran «formed wheret

 

 

A C(>nformal transformation eliminates the e~2' in (6.15) and the motion of 'he confined universe point is described asymptotically by the stationary Hamiltonian

 

 

Now since     's 'he Hamiltonian for a particle moving along the geodesies of a space possessing

Any analytic automorphism of the half-plane P is described by a Möbius bilinear transformation

 

a space-time metric we see that (6.17) shews [70] the Mixmaster evolut'on to be a geodesic flow on a two-dimensional Riemannian manifold (if negative curvature (the Lobatchevsky plane) bounded by the curves (6.14), The two-dimensional metric is just

 

 

This flow can be representid in two ways: If we put sinh 0 = 2r/(l - r) we obtain a hyperbolic flow on the unit disc D = {z: |zj < 1, of constant negative curvature. By a Caytey transformation we can also represent (6.17) as a flow on th? upper hall-plane P = + iy: y > 0} with the Poincare metric

 

 

The set of all such mappings acts transitive y on P and P can be identified with the homogeneous space SL(2 R)/SO(2). The metric (6.19) is also SI (2, R) invariant. Therefore the Mixmaster. which is a set oi

t The negative sign of p) together with the shape of the potential (6.13) cause the universe point to accelcrnte towards regions of h gher fHHenti.il and is m contrast to Hamiltonian flow s in classical mechanics.XD. Bmtem,

discontinuous (broken) geodesies on P is equivalent to t Rtchakm grottp pi]. A fundamental domaint P Can be constructed in P which is a normal polygon bounded by geodesies (i.e. arcs orthogonal to the real

axis).

The description of the Mixmaster as a geodesic flow on the disc D with metric

 

The geodesic flow can be compactified by joining points at infinity and it then becomes a continuous geodesic flow on a compact Riemann surface. The finite measure (area) described by the Poincart metric is just

 

 

was given by Chitre [70]. A fundamental region with a finite number of sides can again be found and the discontinuous geodesic flow on D can be transformed into a continuous geodesic flow on D/F and to this flow the theorems proving ergodicity, mixing and Bernoulli apply. The geodesies on D/F are described by three coordinates; the (x, y) coordinate of points in the fundamental region together with the flow direction, 6, there. The finite measure on the space of y, elements ist

 

 

It is also worth noting that Wheeler's notion [73] of 'superspace' provides the means to establish a close link between the properties of geodesic flows and the dynamics of the Einstein equations. Supcrspace is the space of all three-geometries (positive-definite metrics) and so any space-time which solves Einstein's equations can be represented as a trajectory in superspace. The finite dimensional subset of all homogeneous three-geometries, of which Mixmaster is a member, is called minisuperspace [74]. De Witt [75] and Misner [76] have shown how to place a metric on these spaces and then enables the homogeneous cosmological models to be characterized as geodes:. flows in minisuperspace. The metric underlying the flow (6.19) can be interpreted in this setting.

Finally, we should mention that Bogoiavlenskii and Novikov [77,78] have given a Hamiltonian formuk.tion of the Einstein equations for homogeneous cosmological models. Their Hamiltonian system is an autonomous system of ordinary differential equations; in the Mixmaster case it has six dimensions. To apply standard techniques in the qualitative theory of differential equations it is necessary to compactify the phase space of ihis autonomous system. The compactified dynamical system, S, is equivalent to the original one in its interior and is defined on dS by continuity. The construction of S

• F is i fundamental domain for a discontinuous group G if F is ncn-empty, connected and open, no distinct points z, z' of F are equivalent under G ( c. there is no g 6 G such that g(z) = z') and every point of F is equivalent to some point of F. Every discontinuous group has a (and usually mr ny) fundamental polygon(s).

jTherc also exists a possible connection between Mixmaster descriptions of this sort and the continued fraction description of section 4.4. Artin 172] developed a symbolic description of geodesic flows on the hyperbolic plane by using the continued fraction expansions of the endpoints of the godcsics m the fundamental region. In particular if .*, v € (0,1) have continued fraction expansions x = [ni, n2,...], y = [n, fj,.. .| then x = g(y) for t( e SL<2, R) if and only if there exist k, I such that (-1 )'*1 = / and n*.« = for a Ss 0.must be performed in such a way that the dynamical system on S has a minimum of degenerate singular points. The compact system consists of a collection of séparatrices and so the evolution of the system can be deduced by considering the generic fate of a trajectory passing through the complex of séparatrices. A simple two-dimensional example illustrates the idea [79]: consider the tw.vdimensional autonomous system in the (jc, v) plane described by

 

 

All the critical points (i.e. where the right-hand sides of (6.24) vanish simultaneously) in the phase phre are non-linear and they reside at the points A(-l,-l), B(l, 1), C(l,-1), D(-l, I), E(-2, 1), F(l,2). 0(2,-1), H(-l, -2) and 1(0,0) in the (x-y) plane. The critical points A, B, C, D are saddles with eigenvalues -6 and +2. The critical points E, F, G, H are stable attractors whilst I is neutral. The phase portrait is shown in fig. 5.

The phase space is clearly not compact but it can be made so by a coordinate change to a four dimensional system («, v, w, z) where

 

 

A new time coordinate r can be introduced to simplify the system if we define it by dr d-f ^ The system can be restored to two dimensions by examining its behaviour on the slice where : 0 and w = 0. On this plane we have

 

 

and the new system has the critical points A-I within a bounded circular region as shown in fig. o. These critical points are augmented by J-Q on the ciicular boundaryThe separatrix diagram, fig. 6, indicates the generic fate of any trajectory. For example, a trajectory lying in the cell CADB will wind around in a spiral clockwise, indicative of quasi-periodic, oscillatory behaviour. The motion of a generic trajectory through the cell complex can be determined and a discrete mapping set up to describe the sequence of separatrix changes.