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6.1 Changing coordinates 77 6.2 Rectification of flows 78 6.3 Classical dynamics of collinear he- lium 79 6.4 Rectification of maps 83 6.5 Rectification of a 1-dimensional pe- riodic orbit 85 6.6 Smooth conjugacies and cycle Flo- quet multipliers 86 Summary 86 Further reading 87 Exercises 87 References 88
We owe it to a book to withhold judgment until we reach page
Henrietta McNutt, George Johnson’s seventh-grade English teacher
A Hamiltonian system is said to be ‘integrable’ if one can find a change of coordinates to an action-angle coordinate frame where the phase space dynamics is described by motion on circles, one circle for each degree of freedom. In the same spirit, a natural description of a hyper- bolic, unstable flow would be attained if one found a change of coor- dinates into a frame where the stable/unstable manifolds are straight lines, and the flow is along hyperbolas. Achieving this globally for any- thing but a handful of contrived examples is too much to hope for. Still, as we shall now show, we can make some headway on straightening out the flow locally. Even though such nonlinear coordinate transformations are very im- portant, especially in celestial mechanics, we shall not necessarily use them much in what follows, so you can safely skip this chapter on the first reading. Except, perhaps, you might want to convince yourself that cycle stabilities are indeed metric invariants of flows (Section 6.6), and you might like transformations that turn a Keplerian ellipse into a harmonic oscillator (Example 6.2) and regularize the 2-body Coulomb collisions (Section 6.3) in classical helium.
fast track Chapter 14, p. 201
6.1 Changing coordinates
Problems are handed down to us in many shapes and forms, and they are not always expressed in the most convenient way. In order to sim- plify a given problem, one may stretch, rotate, bend and mix the coor- dinates, but in doing so, the vector field will also change. The vector field lives in a (hyper)plane tangent to state space and changing the co- ordinates of state space affects the coordinates of the tangent space as well, in a way that we will now describe. Denote by h the conjugation function which maps the coordinates of the initial state space M into the reparametrized state space M′^ = h(M), with a point x ∈ M related to a point y ∈ M′^ by
y = h(x) = (y 1 (x), y 2 (x),... , yd(x).
78 CHAPTER 6. GET STRAIGHT
The change of coordinates must be one-to-one and span both M and M′, so given any point y we can go back to x = h−^1 (y). For smooth flows the reparametrized dynamics should support the same number of derivatives as the initial one. If h is a (piecewise) analytic function, we refer to h as a smooth conjugacy. The evolution rule gt(y 0 ) on M′^ can be computed from the evolution rule f t(x 0 ) on M by taking the initial point y 0 ∈ M′, going back to M, evolving, and then mapping the final point x(t) back to M′:
y(t) = gt(y 0 ) = h ◦ f t^ ◦ h−^1 (y 0 ). (6.1)
Here ‘◦’ stands for functional composition h ◦ f (x) = h(f (x)), so (6.1) is a shorthand for y(t) = h(f t(h−^1 (y 0 ))). The vector field x˙ = v(x) in M, locally tangent to the flow ft, is re- lated to the flow by differentiation (2.5) along the trajectory. The vector field y˙ = w(y) in M′, locally tangent to gt^ follows by the chain rule:
w(y) =
dgt dt (y)
t=
d dt
h ◦ f t^ ◦ h−^1 (y)
t= = h′(h−^1 (y))v(h−^1 (y)) = h′(x)v(x). (6.2)
6.1, page 88 With the indices reinstated, this stands for
wi(y) = ∂hi(x) ∂xj
vj (x) , yi = hi(x). (6.3)
Imagine that the state space is a rubber sheet with the flow lines drawn on it. A coordinate change h corresponds to pulling and tug- ging on the rubber sheet smoothly, without cutting, glueing, or self- intersections of the distorted rubber sheet. Trajectories that are closed loops in M will remain closed loops in the new manifold M′, but their shapes will change. Globally h deforms the rubber sheet in a highly nonlinear manner, but locally it simply rescales and shears the tangent field by ∂j hi, hence the simple transformation law (6.2) for the velocity fields. The time itself is a parametrization of points along flow lines, and it can also be reparametrized, s = s(t), with the attendent modification of (6.2). An example is the 2-body collision regularization of the helium Hamiltonian (7.6), to be undertaken in Section 6.3 below.
6.2 Rectification of flows
A profitable way to exploit invariance of dynamics under smooth con- jugacies is to use it to pick out the simplest possible representative of an equivalence class. In general and globally these are just words, as we have no clue how to pick such ‘canonical’ representative, but for smooth flows we can always do it locally and for sufficiently short time, by appealing to the rectification theorem , a fundamental theorem of ordi- nary differential equations. The theorem assures us that there exists a conjug - 15aug2006 ChaosBook.org version11.9.2, Aug 21 2007
80 CHAPTER 6. GET STRAIGHT
learned so far in solving real life physical problems, good news are here. We will apply here concepts of nonlinear dynamics to nothing less than the helium, a dreaded three-body Coulomb problem. Can we really jump from three static disks directly to three charged particles moving under the influence of their mutually attracting or re- pelling forces? It turns out, we can, but we have to do it with care. The full problem is indeed not accessible in all its detail, but we are able to analyze a somewhat simpler subsystem–collinear helium. This system plays an important role in the classical and quantum dynamics of the full three-body problem.
e
He
r (^2)
r (^1)
e
Fig. 6.1 Coordinates for the helium three body problem in the plane.
The classical helium system consists of two electrons of mass me and charge −e moving about a positively charged nucleus of mass mhe and charge +2e. The helium electron-nucleus mass ratio mhe/me = 1836^ is so large that we may work in the infinite nucleus mass approximation mhe = ∞, fixing the nucleus at the origin. Finite nucleus mass effects can be taken into account without any substantial difficulty. We are now left with two electrons moving in three spatial dimensions around the ori- gin. The total angular momentum of the combined electron system is still conserved. In the special case of angular momentum L = 0, the electrons move in a fixed plane containing the nucleus. The three body problem can then be written in terms of three independent coordinates only, the electron-nucleus distances r 1 and r 2 and the inter-electron an- ?? , page ?? gle Θ, see Fig. 6.1. This looks like something we can lay our hands on; the problem has been reduced to three degrees of freedom, six phase space coordinates in all, and the total energy is conserved. But let us go one step further; the electrons are attracted by the nucleus but repelled by each other. They will tend to stay as far away from each other as possible, prefer- ably on opposite sides of the nucleus. It is thus worth having a closer look at the situation where the three particles are all on a line with the nucleus being somewhere between the two electrons. If we, in addi- tion, let the electrons have momenta pointing towards the nucleus as in Fig. 6.2, then there is no force acting on the electrons perpendicular to the common interparticle axis. That is, if we start the classical system on the dynamical subspace Θ = π, (^) dtd Θ = 0, the three particles will remain in this collinear configuration for all times.
He
++
e e
r r
1 2
Fig. 6.2 Collinear helium, with the two electrons on opposite sides of the nu- cleus.
6.3.1 Scaling
In what follows we will restrict the dynamics to this collinear subspace. It is a system of two degrees of freedom with the Hamiltonian
H =
2 me
p^21 + p^22
2 e^2 r 1
2 e^2 r 2
e^2 r 1 + r 2
= E , (6.9)
where E is the total energy. As the dynamics is restricted to the fixed energy shell, the four phase space coordinates are not independent; the energy shell dependence can be made explicit by writing (r 1 , r 2 , p 1 , p 2 ) → conjug - 15aug2006 ChaosBook.org version11.9.2, Aug 21 2007
6.3. CLASSICAL DYNAMICS OF COLLINEAR HELIUM 81
(r 1 (E), r 2 (E), p 1 (E), p 2 (E)). We will first consider the dependence of the dynamics on the energy E. A simple analysis of potential versus kinetic energy tells us that if the energy is positive both electrons can escape to ri → ∞, i = 1, 2. More interestingly, a single electron can still escape even if E is nega- tive, carrying away an unlimited amount of kinetic energy, as the total energy of the remaining inner electron has no lower bound. Not only that, but one electron will escape eventually for almost all starting con- ditions. The overall dynamics thus depends critically on whether E > 0 or E < 0. But how does the dynamics change otherwise with varying energy? Fortunately, not at all. Helium dynamics remains invariant under a change of energy up to a simple scaling transformation; a so- lution of the equations of motion at a fixed energy E 0 = − 1 can be transformed into a solution at an arbitrary energy E < 0 by scaling the coordinates as
ri(E) = e^2 (−E)
ri, pi(E) =
−meE pi, i = 1, 2 ,
together with a time transformation t(E) = e^2 m^1 e/ 2 (−E)−^3 /^2 t. We in- clude the electron mass and charge in the scaling transformation in or- der to obtain a non–dimensionalized Hamiltonian of the form
H = p^21 2
p^22 2
r 1
r 2
r 1 + r 2
The case of negative energies chosen here is the most interesting one for us. It exhibits chaos, unstable periodic orbits and is responsible for the bound states and resonances of the quantum problem treated in Section ??.
6.3.2 Regularization of two–body collisions
Next, we have a closer look at the singularities in the Hamiltonian (6.10). Whenever two bodies come close to each other, accelerations become large, numerical routines require lots of small steps, and nu- merical precision suffers. No numerical routine will get us through the singularity itself, and in collinear helium electrons have no option but to collide with the nucleus. Hence a regularization of the differential equations of motions is a necessary prerequisite to any numerical work on such problems, both in celestial mechanics (where a spaceship exe- cutes close approaches both at the start and its destiantion) and in quan- tum mechanics (where much of semiclassical physics is dominated by returning classical orbits that probe the quantum wave function at the nucleus). There is a fundamental difference between two–body collisions r 1 = 0 or r 2 = 0, and the triple collision r 1 = r 2 = 0. Two–body collisions can be regularized, with the singularities in equations of motion removed by a suitable coordinate transformation together with a time transfor- mation preserving the Hamiltonian structure of the equations. Such ChaosBook.org version11.9.2, Aug 21 2007 conjug - 15aug
6.4. RECTIFICATION OF MAPS 83
We now apply this method to collinear helium. The basic idea is that one seeks a higher-dimensional generalization of the ‘square root removal’ trick (6.14), by introducing a new vector Q with property r = |Q|^2. In this simple 1-dimensional example the KS transformation can be implemented by
r 1 = Q^21 , r 2 = Q^22 , p 1 =
P 1
2 Q 1
, p 2 =
P 2
2 Q 2
and reparametrization of time by dτ = dt/r 1 r 2. The singular behav- ior in the original momenta at r ?? ,^ page^ ?? 1 or^ r 2 = 0^ is again compensated by stretching the time scale at these points. The Hamiltonian structure of the equations of motions with respect to the new time τ is conserved, if we consider the Hamiltonian
Hko =
(Q^22 P 1 2 + Q^21 P 22 ) − 2 R^212 + Q^21 Q^22 (−E + 1/R^212 ) = 0 (6.17)
with R 12 = (Q^21 + Q^22 )^1 /^2 , and we will take E = − 1 in what follows. The equations of motion now have the form
P^ ˙ 1 = 2Q 1
[
P 22
− Q^22
Q^22
R^412
)]
; Q˙ 1 =
P 1 Q^22 (6.18)
P^ ˙ 2 = 2Q 2
[
P 12
− Q^21
Q^21
R^412
)]
; Q˙ 2 =
P 2 Q^21.
Individual electron–nucleus collisions at r 1 = Q^21 = 0 or r 2 = Q^22 = 0 no longer pose a problem to a numerical integration routine. The equations (6.18) are singular only at the triple collision R 12 = 0, i.e., when both electrons hit the nucleus at the same time. The new coordinates and the Hamiltonian (6.17) are very useful when calculating trajectories for collinear helium; they are, however, less in- tuitive as a visualization of the three-body dynamics. We will therefore refer to the old coordinates r 1 , r 2 when discussing the dynamics and the periodic orbits. To summarize, we have brought a 3-body problem into a form where the 2-body collisions have been transformed away, and the phase space trajectories computable numerically. To appreciate the full beauty of what has been attained, you have to fast-forward to Chapter ?? ; we are already ‘almost’ ready to quantize helium by semiclassical methods.
fast track Chapter 5, p. 69
6.4 Rectification of maps
In Section 6.2 we had argued that nonlinear coordinate transformations can be profitably employed to simplify the representation of a flow. We shall now apply the same idea to nonlinear maps, and determine a smooth nonlinear change of coordinates that flattens out the vicinity of ChaosBook.org version11.9.2, Aug 21 2007 conjug - 15aug
84 CHAPTER 6. GET STRAIGHT
Fig. 6.3 (a) A typical trajectory in the [r 1 , r 2 ] plane; the trajectory enters here along the r 1 axis and escapes to infinity along the r 2 axis; (b) Poincar´e map (r 2 =0) for collinear helium. Strong chaos pre- vails for small r 1 near the nucleus.
-0.
-0.
-0.
-0.
0
(^01 2 3 4 5 6 7 8 9 )
2
4
6
8
10
0 2 4 6 8 10
a)
r 1
b)
2 p 1
r 1
r
a fixed point and makes the map linear in an open neighborhood. In its simplest form the idea can be implemented only for an isolated nonde- generate fixed point (otherwise are needed in the normal form expan- sion around the point), and only in a finite neigborhood of a point, as the conjugating function in general has a finite radius of convergence. In Section 6.5 we will extend the method to periodic orbits.
6.4.1 Rectification of a fixed point in one dimension
6.2, page 88 Consider a 1-dimensional map xn+1 = f (xn) with a fixed point at x = 0 , with stability Λ = f′(0). If |Λ| = 1, one can determine term-by-term the power series for a smooth conjugation h(x) centered at the fixed point, h(0) = 0, that flattens out the neighborhood of the fixed point
f (x) = h−^1 (Λh(x)) (6.19)
and replaces the nonlinear map f (x) by a linear map yn+1 = Λyn. To compute the conjugation h we use the functional equation h−^1 (Λx) = f (h−^1 (x)) and the expansions
f (x) = Λx + x^2 f 2 + x^3 f 3 +... h−^1 (x) = x + x^2 h 2 + x^3 h 3 +.... (6.20)
Equating the coefficients of xk^ on both sides of the functional equation yields hk order by order as a function of f 2 , f 3 ,.. .. If h(x) is a conjuga- tion, so is any scaling h(bx) of the function for a real number b. Hence the value of h′(0) is not determined by the functional equation (6.19); it is convenient to set h′(0) = 1. The algebra is not particularly illuminating and best left to comput- ers. In any case, for the time being we will not use much beyond the first, linear term in these expansions. Here we have assumed Λ = 1. If the fixed point has first k− 1 deriva- tives vanishing, the conjugacy is to the kth normal form. In several dimensions, Λ is replaced by the Jacobian matrix, and one has to check that the eigenvalues M are non-resonant, that is, there is no integer linear relation between the stability exponents (5.4). conjug - 15aug2006 ChaosBook.org version11.9.2, Aug 21 2007
86 CHAPTER 6. GET STRAIGHT
6.6 Smooth conjugacies and cycle Floquet
multipliers
In Section 5.2 we have established that for a given flow the cycle Flo- quet multipliers are intrinsic to a given cycle, independent of the staring point along the cycle. Now we can prove a much stronger statement; cycle Floquet multipliers are metric invariants of the flow, the same in any representation of the dynamical system. That the cycle Floquet multipliers are an invariant property of the given dynamical system follows from elementary considerations of Sec- tion 6.1: If the same dynamics is given by a map f in x coordinates, and a map g in the y = h(x) coordinates, then f and g (or any other good representation) are related by (6.2), a reparametrization and a coordi- nate transformation g = h ◦ f ◦ h−^1. As both f and g are arbitrary representations of the dynamical system, the explicit form of the conju- gacy h is of no interest, only the properties invariant under any trans- formation h are of general import. Furthermore, a good representation should not mutilate the data; h must be a smooth conjugacy which maps nearby cycle points of f into nearby cycle points of g. This smoothness guarantees that the cycles are not only topological invariants, but that their linearized neighborhoods are also metrically invariant. For a fixed point f (x) = x of a 1-dimensional map this follows from the chain rule for derivatives,
g′(y) = h′(f ◦ h−^1 (y))f ′(h−^1 (y))
h′(x) = h′(x)f ′(x)
h′(x)
= f ′(x) , (6.23)
and the generalization to the Floquet multipliers of periodic orbits of d-dimensional flows is immediate. As stability of a flow can always be rewritten as stability of a Poincar´e section return map, we find that a Floquet multiplier of any cycle, for a flow or a map in arbitrary dimension, is a metric invariant of the 6.2, page 88 dynamical system.
in depth: Appendix B.1, p. 291
Summary
Dynamics (M, f ) is invariant under the group of all smooth conjuga- cies (M, f ) → (M′, g) = (h(M), h ◦ f ◦ h−^1 ). This invariance can be used to (i) find a simplified representation for the flow and (ii) identify a set of invariants, numbers computed within a particular choice of (M, f ), but invariant under all M → h(M) smooth conjugacies. conjug - 15aug2006 ChaosBook.org version11.9.2, Aug 21 2007
Further reading 87
The 2 D-dimensional phase space of an integrable Hamiltonian sys- tem of D degrees of freedom is fully foliated by D-tori. In the same spirit, for a uniformly hyperbolic, chaotic dynamical system one would like to change into a coordinate frame where the stable/unstable man- ifolds form a set of transversally intersecting hyper-planes, with the flow everywhere locally hyperbolic. That cannot be achieved in gen- eral: Fully globally integrable and fully globally chaotic flows are a very small subset of all possible flows, a ‘set of measure zero’ in the world of all dynamical systems. What we really care about is developping invariant notions of what a given dynamical system is. The totality of smooth one-to-one nonlin- ear coordinate transformations h which map all trajectories of a given dynamical system (M, f t)^ onto all trajectories of dynamical systems (M′, gt) gives us a huge equivalence class, much larger than the equiv- alence classes familiar from the theory of linear transformations, such as the rotation group O(d) or the Galilean group of all rotations and translations in Rd. In the theory of Lie groups, the full invariant speci- fication of an object is given by a finite set of Casimir invariants. What a good full set of invariants for a group of general nonlinear smooth conjugacies might be is not known, but the set of all periodic orbits and their stability eigenvalues will turn out to be a good start.
Further reading
Rectification of flows. See Section 2.2.5 of Ref. [12] for a pedagogical introduction to smooth coordinate reparametrizations. Explicit examples of transformations into cannonical coordinates for a group of scalings and a group of rotations are worked out. Rectification of maps. The methods outlined above are standard in the analysis of fixed points and construc- tion of normal forms for bifurcations, see for example
Ref. [21, 2, 4–9, 9]. The geometry underlying such methods is pretty, and we enjoyed reading, for example, Percival and Richards [10], chaps. 2 and 4 of Ozorio de Almeida’s monograph [11], and, as always, Arnol’d [1]. Recursive formulas for evaluation of derivatives needed to evaluate (6.20) are given, for example, in Ap- pendix A of Ref. [5].
Exercises
(6.1) Coordinate transformations. Changing co- ordinates is conceptually simple, but can become confusing when carried out in detail. The difficulty arises from confusing functional relationships, such as x(t) = h−^1 (y(t)) with numerical relationships,
such as w(y) = h′(x)v(x). Working through an ex- ample will clear this up. (a) The differential equation in the M space is x ˙ = { 2 x 1 , x 2 } and the change of coordinates from M to M′^ is h(x 1 , x 2 ) = { 2 x 1 + x 2 , x 1 − ChaosBook.org version11.9.2, Aug 21 2007 exerConjug - 15feb
