Abstract Topoktgica! and numerical techniques are used to show that the problem of trapped charged particle motion in a magnetic dipoie field is insoluble. Similar results hold for motion in the earth's magnetic field and are of interest for radiation belt phenomena. Pedagogical discussion is devoted to the subject of how it can happen that a classical mechanics problem is insoluble and in what sense. It is shown that the com plete adiabatic magnetic moment series is divergent and that due to the existence of homoclinic points the solutionis to the equations of motion are too complicated to be written in closed form. As a consequence, ites is currently no rigorous theoretical explanation for the empirical success of adiabatic orbit theory, and a completely satisfactory mathematical justification will be far from easy. 1. Introduction * , cs-v-isttal ingredient for an understanding of magneto-■.:'c arftd r?diation belt phenomena is a knowledge of the ' o* .barged particles trapped in the earth's magnetic Tr^pned orbits can be approximately described in terms c -v -ce.-,--magnetic moment, longitudinal, and flux inn:s ^Northrop, 1963; Northrop and Teller, I960] . These :--.ita5 are not true invariants because they are not actual .im.\vcf motion over the course of an orbit However, they dufhatically invariant* in that they remain approximately m.T,\: for moderate lengths of time for low-energy orbits. ;4?ksnt!y, they can be used to predict particle motion ..\ *V moderately short times. In actual practice it is _-,. .d>> tacitly assumed that the adiabatic invariant .'imm hold good for arbitrarily long times. Such an as-* ■ ; satis may be correct, but its use certainly requires discusi^& eventual mathematical justification. \w of the uncertainties associated with the adiabatic --''■an&>predi'Uions, it would be highly desirable to have an :v*>, stable model problem which in some sense approxi-\ tterirue problem. One obvious possibility is to replace ■•■ -rtV* .actual magnetic field by that ofa perfect dipole. We ■•:_i* ahe problem of determining the orbits in a purê ';-*--';iwfpole field the Stormer problem in honor oiStirmer •; "j., vh&€> first considered it [Rossi and Olbert, 1970] . If the ""ier problem could be solved exactly, we could hope to --th^Ml problem by perturbation methods. ; r thelbasis of past experience in other areas of physics, we , i oj^kely hope that the simplifications and symmetry in--ucesf. i>y a pure dipole field do indeed lead to an exactly -e p^hlem. For example, the motion of a satellite about v-uriifc becomes the exactly soluble Kepler problem if we -.dx #*e earth's gravitational field by ignoring quadrupole : ;gta+rnornent terms. However, it has been well known to v'a! mechanicians since the time of Pioncare [1892] and '«"* l®PMttaker, 1937] that 'most' classical mechanics prob-888a:dbb7:2d49:b3f6:8605:bf4e:caed:e5a4* r&: insoluble.' The purpose of this paper is to show that -^^^m^r problem for trapped orbits belongs to this inlt y^ority. We hope that our warning will spare aspiring -'•---iia. students and others the expense of spending long -:'-^.nta>2Fs in the hope that to the amazement of all, they ?"•' $m®ver just the right canonical transformation which Si*v i2i pbsma Physics Laboratory, Princeton University, Prince-*** Jersey 08540°? } ipkt © 1976 by the American Geophysical Union. leads to an exact solution and lasting fame. At the same time. we hope that our discussion will prove instructive as to why and in what sense a classical mechanics problem can be in soluble, since this subject is not common knowledge among physicists and is rarely touched upon in mechanics courses or texts. Finally, of more interest from a practical point of view. we will show that insolubility implies that the complete adia batic magnetic moment series diverges. Consea^uenlly^jhere is aX^present_nq^rigorous mathematical ex^TanatlorL fojLjhe of adiabatic success ma^netospheric: pKenomo____a n d coTreiati ngsp_acecra ft_j ata. Rather, its success musj^ currently tJeview^ijU^^â T^pIajisTbkjyinor rrmEdT. "^Wenave tried to make our exposition as simple and non technical as possible. More detailed calculations will be pre sented elsewhere. Section 2 describes briefly the equations of motion for the Stormer problem and the use of dipolar coordi nates. In section 3 we convert the Stormer problem into the determination of an area preserving map M. Section 4 de scribes various properties of area-preserving mappings includ ing the possible existence of a 'homoclinic' point. Section 5 shows that if M has a homoclinic point, then the Stormer problem is insoluble. Here we also discuss what 'insolubilin" means. We show in section 6 by direct numerical integration that the map M has homoclinic points. Our results and the current state of the Stormer problem are summarized in a final section. Equations of Motion and Coordinates A portion of a typical trapped St0rmer orbit is illustrated in Figure 1 . For a nonrelativistic particle of charge q and mass m the orbit is generated by the Hamiltonian H = 0 w ) { p . f + p p * -f \ i p j p ) -q A t f ) ( I ) where p, z, 4> are cylindrical coordinates and the vector poten tial A describing the dipole field is given by A = 0£Hlpr Equations of Motion and Coordinates (2) with r2 = p2 + r2. Here SfTt is the magnitude of the dipole moment. (The relativistic case is also described by (1) if jw is replaced by 7m.) The motion consists of three pans: a gyration about a field line, a bouncing back and forth across the equa torial plane along the line between mirror points to form a kind of spiral, and a slow drift about the earth. For a fuller discussion of the motion and a fuller exposition of some of the. points which are to follow, we refer the reader to an eariier article [Dragt, 1965] . Inspection of (1) and (2) shows that H is independent of </> as we might well expect from the axial symmetry of the problem. It follows that p* = -eH/c<t> = 0 Equations of Motion and Coordinates and hence p^ is a constant of motion. It can conveniently be written in the form p* = qNiT (4) where T is an integration constant having the dimensions of a reciprocal length. An analysis of the properties of.// shows that trapped orbits can occur only if T > 0 and that in this case, particles with sufficiently low energy spiral about a field line satisfying r = I*"1 cos2 A Equations of Motion and Coordinates where, as, \ft Figure 1 , X denotes geomagnetic latitude. Equations of Motion and Coordinates Applying Hamilton's equations we find 0 = c H / e p 0 = ( / > * -q p A t ) / ( m p 2 ) ( 6 ) Equations of Motion and Coordinates As far as the motion in p and z is concerned, we note that we may replace p* in H by its value given in (4). After this substitution is made, we can regard H(ppp2\ pz\p$ = q&lT) as a deduced' Hamiltonian describing two-dimensional motion in the p, z plane. Once this motion is determined to give p(t) and r(/), we can easily determine <t>{t) simply by integrating (6): Equations of Motion and Coordinates (r) _= f dt (gMT -qpA^/imp2) Equations of Motion and Coordinates Therefore in what follows we shall concentrate on finding p(t) and z(f). At this point it is convenient to introduce dimensionless space and time variables p\ r\ t' by the rules z' = zT (8) p' = pr t' = tT3qm/m In these variables the trapped particle gyrates about a guiding field line obeying r' = cos2 X (9) Equations of Motion and Coordinates Also the particle has unit cyclotron frequency when it is in the equatorial plane. The motion is governed by the dimensionless Hamiltonian H = -Kp,2 + Pp2) + v(p9z) 00)^( Equations of Motion and Coordinates For notational convenience we have omitted the primes in \\. and (11), and we shall continue to do so henceforth. All fy^. references w ill be to dimensionless variables and dimension equations of motion. We see from (10) that the motion in the p, z plane is \i same as that of an imaginary particle of unit mass moving j the effective two-dimensional potential K(p, z). Therefore %» can get a qualitative picture of possible orbits by examining i Figure 2 usually called the thalweg (German: 'valley way'), and ispo> tive elsewhere. (Note that (12) is the same as the guiding Ik line (9)). Also the walls of the potential become steeper as on proceeds from the equator, z = 0, down the thalweg tow&r. either pole. Finally, there is a pass at z = 0, p = 2, where Vh& the value -fc. From energy conservation we conclude that all orbits wbk begin in the valley and which satisfy W02 < fa where WVb" dimensionless 'velocity' defined by W* = 1H W cannot escape to infinity. These are the orbits which we ha. called 'trapped.' Figure 3 shows a typical portion of a trapped orbit in thc|. plane obtained by numerical integration. The motion consk of oscillations about the thalweg superimposed upon meu along the^thalweg. Because the walls of the potential beco. steeper^!? one proceeds down the thalweg, our imagine particle experiences a retarding force. Consequently, theorreventually turns around. The oscillations about the thai*? correspond to gyrations about the guiding treld line in the V, three-dimensional orbit, and the motion back and forth -li the thalweg corresponds to the bouncing motion between iru'-ror points. Equations of Motion and Coordinates Since in first approximation the motion in the p, z pk? consists of oscillations about the thalweg superimposed um otion along the thalweg, it is convenient to take this feaur into account by the introduction of orthogonal 'dipol coordinates qx and q2. Equations of Motion and Coordinates ,pes o£ constant ^2 correspond to dipole field lines. In par=vi.lar, litieiine ^2 = 0 corresponds to the thalweg. Lines of v.stant:#, are orthogonal to lines of constant q2. Figure 4 v ;v us ita-ao-rbit of Figure 3 as it appears in terms of dipolar >-!v.rdinaus&. The motion has now been separated out, in first ; proxien_ftion, into oscillations about q2 = 0 superimposed p m motuon along the qx axis. We cta^this section by noting the form of the Hamiltonian dipolan coordinates. Letp1 andp2 be momenta canonically ,..:!;jUgatfe_Kr-^-and0a.Then after calculation one finds that the ti.niltomi&n is given (in mixed variables) by / / -i l f t f t / A i ) 1 + ( P 2 / h 2 f + r \ q 2 + 1 ) " V l ( 1 5 ) v re re h2 -rV{r2 + 3z2) V = pV4/(^2 + 3z2) (16) (17) partiaiilar, for future reference we observe that H for pped orfMits can be expanded in a power series in the q's and of theHonn [Contopoulos and Vlahos, 1975] Equations of Motion and Coordinates 2 + 21g.2p12 Equations of Motion and Coordinates + 10?24 + 39lV + VPif) (21) Equations of Motion and Coordinates 1. Conversion into a Mapping Problem ' he pur^M&se of this section is lo explain how the particle •ion des__?ibed in the previous section can be used to genera mafgfsaag M. Our method involves the use of what is -d a _ug$_ace of section and dates back to Poincare's celeed wgi& mn celestial mechanics. onsiderr_2b_ four-dimensional phase space consisting of the ables #.";£, pp, pz. From Hamiltonian mechanics v%e know every-,<s£&il'm the configuration space p, z corresponds to a ectory imtphase space. Furthermore, there is a unique traory tbmtEggh each point in phase space, and trajectories -r interss&gt unless they happen to close on themselves. Since WV = 2H is a constant of motion, it is convenient to group together all trajectories with the same WV-They evidently lie on a three-dimensional hypersurface. Now con sider all trajectories which correspond to trapped orbits in configuration space and which have a fixed value of WV-From our discussion in section 2 of the nature of orbits, st is in tuitively obvious that each such phase space trajectory, when it is extended far enough forward or backward in time, must cross the hyperplane z = 0. That is, all orbits must cross the equatorial plane at least once. A rigorous proof has been given by DeVogelaere [1954] . Put another way, the 'surface* z = 0 cuts across every (trapped) trajectory and thus may be called a surface of section. Equations of Motion and Coordinates Let us record the values of p, p = pp, and z = pz at the moment of crossing. Their values and the equations of motion derived from H allow a complete reconstruction of the w hole trajectory. Futhermore, since we have agreed to fix WV, the value of z is redundant because by solving (13)for z when z = 0 we find Equations of Motion and Coordinates Thus a trajectory is specified by the two numbers p\Zm0 and p\2m-o. and the surface of section is effectively two-dimensional. There is, of course, an ambiguity in the sign of z as given by (22). So we should really say that each p, p pair in general specifies two trajectories, one with z|z.0 > 0 and one with z|*"0 < 0. However, we observe that V is symmetric about the p a.xis: Equations of Motion and Coordinates ] is an orbit, so is its mirror image [p(/) = fit), z{t) = ~g(t)). Thus the two possible choices of sign for z amount to choices between two mirror images. Equations of Motion and Coordinates We also need to make one further clarification. It is obvious from (11) and (22) that only a certain region of the p, p plane leads to real values of z and that outside this region, z is pure imaginary. The region of real z, which we shall call the phys ical region, is the boundary and interior of the curve Equations of Motion and Coordinates The boundary itself gives vanishing values ofz, and therefore all points on the boundary are points of that orbit which is confined to the p axis in configuration space. (This orbit cor responds to an equatorial orbit in the p, z, (f> space.) Points inside the boundary correspond to orbits which leave the p axis and extend into either the northern or southern hemi spheres. Equations of Motion and Coordinates We have seen that a p, p pair in the physical region in _330 Dragt and Finn: Insolubility of Orbits in a Dipole Field general specifies two orbits. We shall see now how the equa tions of motion can be used to generate a mapping M of the physical region into itself [DeVogelaere, 1958; Godart, 1970] : Select a p, p point in the interior of the physical region, and compute z by using (22) with a positive square rooi. Employ ing these values (plus z = 0) as initial conditions, compute an orbit. Analysis shows that this orbit will either go to the origin of the dipole by way of the northern hemisphere or ii will turn around and recross the p axis; i.e., z = 0. If the trajectory recrosses the p axis, record the crossing values p\ p. Now if the orbit specified by the initial values p, p does not go to the origin, then we may define a mapping M by the rule Equations of Motion and Coordinates For example, the orbit in Figure 3 was launched with WV = 0.005, z = 0, p = 1.070, p = 0 which, by using (22) To continue our discussion, we use the fact that the orbit to the origin by way of each hemisphere is unique [Braun, 3970a] . Therefore M has been defined everywhere within the interior of the physical region except for one point. Let O be the p, p point which generates the orbit to the origin. This trajectory never returns. However, we can extend the definition of M to O by defining the return orbit to be the outgoing orbit retraced backwards. Thus if O has coordinates pt. and p0. the point MO has coordinates p0 and -p0. It can be shown that this extension preserves continuity. Finally, we extend the definition of M to points on the boundary of the physical region by again in voking continuity. In conclusion, we have defined a mapping M of the entire physical region into itself. In particular, the boundary is mapped into itself, and the interior is mapped into itself. Equations of Motion and Coordinates In defining the mapping M from p, p to p , p' we used the positive square root in (22). Let us call the orbit thus generated [p+(/). z+(r)]. If tR is the time at which the orbit again returns to cross the p axis, we have *+('*) = 0 Suppose we had used the negative sign instead. Then we wquv have generated the orbit [p_(f), *_(f)], where according to 0JI earlier discussion, Equations of Motion and Coordinates In particular, at / = tR we find z-Ur) = -z+(tR) = 0 P-(tR) = p+(tR) = p' Equations of Motion and Coordinates Thus the definition of M is independent of the sign of the square root. What is M good for? In brief, a knowledge of M is equiv*. lent to a knowledge of the long-time behavior of orbits. Fasuppose, as in Figure 5 , we launch an orbit from some point f in the physical region into, say, the northern hemisphere (z v 0). The orbit will return through the surface of section at tk point MP. It will then continue on into the southern heinisphere only to return again through the point MMP. Funk: successive crossings generate the points M3P, M4P, etc. Thus* z description of the behavior of Mn for large n is equivalent to* description of the long-time behavior of all orbits. PiK&^ties OF Area-Preserving Maps The mapping M has three simple properties which folio* almost directly from its definition and which also serve ?.■ starting points for deeper investigation. PiK&^ties OF Area-Preserving Maps 1. M has an inverse M~l. M also is continuous and t\'.fki> entiable. That is, if P is some point in the p, p plane and Q *■ MP is its image under M, then small changes in P result sr small changes in Q. In fact, if P is not the point O which lead to the origin, then the coordinates of Q = MP can be differcr. tiated with respect to the coordinates of P. These assertion* follow from the fact that the solution to a differential equatior depends continuously and differentiably on the initial coni tions provided the solution does not pass through a poir (such as the origin) where the differential equation coiitDir singular terms. The existence of the inverse mapping M" follows from the observation that an orbit can always be traced backwards in time. PiK&^ties OF Area-Preserving Maps 2. M is area preserving. This means that if R is a region ir the p, p plane and if/?' is its image under the action of M, the" R Bind R' have the same area. The result is a consequence of ifc fact that the equations of motion are derivable from a Mara* iltonian [ArnoldandAvez, 1968] . PiK&^ties OF Area-Preserving Maps 3. M has fixed points. A fixed point is a point which is seninto itself under the action of M. Evidently, fixed points of #• or of powers of Af, correspond to periodic orbits in the p.« plane. The existence of such orbits is intuitively obvious aitf can be proven rigorously [DeVogelaere, 1958] . PiK&^ties OF Area-Preserving Maps Properties (1) and (2) can be combined to give a classitka tion of the fixed points discussed in property (3). Our disc--sion will apply to area-preserving maps in general. To M®u our notation more precise, we shall now denote points in thtPp plane by specifying a two-component vector. Let a be a pî n the physical region and let b be its image under the action & M. We write b = Ma. Next consider the action of M o« *n earby point a + e, where c denotes a small vector. Since V* differentiable, we may expand M(& + <) in a power series in $' components of € to get an expression of the form Dragt and Finn: Insolubility of Orbits in a Dipole Field 23; PiK&^ties OF Area-Preserving Maps -,c:e £_-. denotes a 2 X 2 matrix. We shall call La the linear -oi MM the point a. PiK&^ties OF Area-Preserving Maps Tne sm&tTix La has two important properties. First, it is . ^uslw #eal, since M acts on real vectors to produce real ■v vrs, Secondly, La has unit determinant det La = 1 PiK&^ties OF Area-Preserving Maps -result follows from property (2): Let us consider some ... rt^mm.R centered around a and transform it to a small : n Jt" .centered around b by the application of M. Then the of lite areas of these two regions is given by the Jacobian \f *kmh is just det La. Since M is area preserving, the ratio :e sFCrSasfiiust be 1, and hence (29) follows. \ow suppose a is a fixed point; i.e., M_ = a. Then for any •: .-by pt_&nt'a + € we have PiK&^ties OF Area-Preserving Maps ■^(a ■*-e) = M(a + Lat) + 0(e2) = a + Lfl2€ + 0(62) n gemeral for any power, Mn(2L + €) = a + Ia*« + 0(t2) (30) : ^ee tifo&i in first approximation the behavior of Mn at a v888a:dbb7:2d49:b3f6:8605:bf4e:caed:e5a4 pointe is governed by its linear part. Therefore we should r rewiiat can be said about La and its powers. 1: e bdfaavior of La is characterized by its eigenvectors and 888a:dbb7:2d49:b3f6:8605:bf4e:caed:e5a4-;vai__».*Lct us call the eigenvalues Xx and X2. Then in view "s>)m®:*must have PiK&^ties OF Area-Preserving Maps . jse i_he determinant of a matrix equals the product of its :-.-ivali__5«. Also if Xi happens to be complex, X2 must be its plex conjugate, since La and hence its characteristic equaare vsss&l. Combining these two properties, we see after a -; an_%!6is that there are only five possibilities: PiK&^ties OF Area-Preserving Maps According to the hyperbolic possibility, one eigenvalue, t Xj._$sre_l and greater than 1. Then we have X,-= X and X2 -\ i * , i > 1 . -AoQfcfcOtding to the elliptic possibility, both eigenvalues " .omplfeKiand lie on the unit circle. Then Xj = e** and X2 = wte»^'is some real angle different from zero or a mulAc__$n£ing to the inversion hyperbolic possibility, botĥ 888a:dbb7:2d49:b3f6:8605:bf4e:caed:e5a4n_Jaags -are real and negative. Then X2 = -X and X2 = : A witib& > I. : Acossding to the parabolic possibility, both eigenvalues ---! +1. PiK&^ties OF Area-Preserving Maps Acsgarding to the inversion parabolic possibility, both e^valaig^ equal -1. f" or otar purposes we shall be particularly interested in the ">' poSr^KlIty, the hyperbolic case. PiK&^ties OF Area-Preserving Maps 'jpposBfc is a hyperbolic fixed point. That is, Lh is hyper--iv.. We .mt. e going to explore the effect of powers of M on '••-by psM.fcs.li -f c. In view of (30) we begin by studying the "^is of .t^.on €. Since h is hyperbolic, Xx and X2 are real and 888a:dbb7:2d49:b3f6:8605:bf4e:caed:e5a4-enL Hfeerefore Lh has two real linearly independent ei-:n'*<xUM_s^h'ich we denote by \x and v2. Let us expand c in '-■-> of ^;and v2 by writing Allows gjh&t PiK&^ties OF Area-Preserving Maps We see that if c has components tx and e2, then Lhn€ has components elX1n and €2X2\ In particular, in view of (31), the product of the components of Lhnt is dCa, independent of the value of n. Let us regard the vectors Vj and v2 as a set of (usually) oblique axes. Then the set of points given by (32) with the product €,c2 put equal to a constant is easily recog nized as a hyperbola Jt follows from (33) that the points Lhn* for fixed c and variable n all lie on the same hyperbola and the action of Lh is to move points either along hyperbolas or _k>T»t, the axes yx and v2. PiK&^ties OF Area-Preserving Maps The case of points on the axes is particularly simple. Then we have either e2 = 0 or €, = 0 and hence either PiK&^ties OF Area-Preserving Maps respectively. Here we have used our convention Xj = X > 1 and X2 = 1/X. We see that the action of Lh is to move points on v, along Vi away from the origin and points on v2 along v2 into the origin. Figure 6 illustrates the action of Lh both for this case and for points off the axes. PiK&^ties OF Area-Preserving Maps This completes our exploration of Lh and its action. Return ing to (30), we see that 'in the small/ when higher powers in « are neglected, the effect of M itself is to move points near h along on hyperbolas or their asymptotes, the axes. Another way of describing this situation is to say that neglecting higherorder terms in €, there is a set of curves near h, namely, hyperbolas and their axes, which are each invariant under M. PiK&^ties OF Area-Preserving Maps It is a remarkable result that the full map M, with no powers oft neglected, also possesses invariant curves in the neighbor hood of a hyperbolic fixed point. Naturally enough, near the fixed point these curves look like the hyperbolas and their asymptotes that we obtained by examining Lh. The existence of invariant curves which pass through the fixed point which are the analogs of the v, and v2 axes was proved by Hadamard [1901] . Formal series expressions for these curves and the invariant curves analogous to hyperbolas were obtained by Birkhoff [1920] . Finally, Moser [1956] established that these series actually converge near the hyperbolic fixed point and thus proved the existence in general of invariant curves near hyperbolic fixed points. Fig. 6 . The action of Lh illustrating that points on vx are moved outward, those on v2 are moved inward, and others are moved on hyperbolas. PiK&^ties OF Area-Preserving Maps Let us now focus our attention on two particular invariant curves of M. namely, those which pass through the fixed point h itself. As was mentioned earlier, these two curves are the analogs of the v, and v2 axes that we found when we neglected the higher-order terms in c. By (30) they are tangent to v, and v2, respectively, as they pass through h. We shall call the curve which is analogous to the v, axis the unstable manifold and denote it by the symbol Wu. Similarly, we will denote by W8 the curve w hich is analogous to the v2 axis and call it the stable manifold. They are given these names because in analogy to (34) it can be shown that they have the properties lim Af "p = h n-*cD PiK&^ties OF Area-Preserving Maps That is, W8 consists of all points p which are ultimately moved into h under repeated action of M. Hence we get the name stable. Note that by (346), points on the v2 axis have this property if we consider only the action of Lh. By contrast, Wu consists of all points which go into h under repeated action of M'\ and hence they are moved away from h under repeated action of M. Evidently, from (34a). this is analogous to the behavior of points on the \, axis under the repeated action of Lh~l and L*. PiK&^ties OF Area-Preserving Maps To get an idea of how our discussion works out in a specific case, let us consider what might be regarded as the simplest nonlinear area-preserving map. It is a map consisting of just linear and quadratic terms. If we select x and y as Cartesian coordinates in the plane, our simple example is given by the rule PiK&^ties OF Area-Preserving Maps where, for the moment, X is an adjustable parameter. We shall denote this mapping by the symbol Mc in honor of Cremona, who was an early student of polynomial maps. PiK&^ties OF Area-Preserving Maps It is easily verified that Lxy has determinant +1, so our ex ample is indeed area preserving as advertised. PiK&^ties OF Area-Preserving Maps Next we observe that Mc has the origin as a fixed point. That is, the point a* = y = 0 is sent into itself. Moreover, at the origin, Lxy has the forrn^% 0 0 (38) which shows that X and X-1 are the eigenvalues associated with the fixed point. Therefore if we take for X some number greater than 1, the origin is a hyperbolic fixed point. Finally, in this case the eigenvectors v, and v2 lie along the x and y axes, respectively. We therefore expect that the unstable manifold Wu for Mc will be tangent to the x axis at the origin and the stable manifold Ws will be tangent to the y axis. Figure 7 shows various invariant curves, including Wu and W8. in the neighborhood of the origin for Mc in the case X -3. These curves were obtained by analytical and numerical means which we will publish in detail elsewhere [Finn, 1974] . We see that they indeed have the expected hyperbolic structure near the origin. PiK&^ties OF Area-Preserving Maps The stage is now set for us to make a fundamental observa tion first about Mc and then the general case. Suppose we apply successive powers of Mc and Mc~* to the invariant curves that we have obtained near the origin. This operation will have the effect of extending the curves away from the origin. Moreover, the extended curves will still be invariant, for by construction if p is a given point on an original curve or its extension, Mcp and Mc~xp will also be on the curve or its e x t e n s i o n . PiK&^ties OF Area-Preserving Maps ; ' ■ : ■■■ In particular, let us extend in this fashion the unstable and stable manifolds, Wu and W8. Figure 8 shows the result ob tained numerically for X = 3. We see that the piece of Wu corresponding to positive jc and the piece of W* corresponding to negative y extend off to infinity. However, the other two pieces of Wu and Ws approach each other and eventually meet. Indeed, they cross each other with a nonzero angle. PiK&^ties OF Area-Preserving Maps The possibility that the Wu and W8 emanating from a hyper bolic fixed point h of an area-preserving map might intersect with a nonzero angle (rather than joining smoothly) was first * -. envisioned by Poincare [1892] . He called such a point of inter sections a homoclinic point and showed that if a homoclinic point existed, Wu and W8 must have an extremely complicated oscillatory structure. PiK&^ties OF Area-Preserving Maps Poiacare reasoned roughly as follows: Let us denote a homoclinic point of an area-preserving map by the symbol K. Now arpply powers of the map and its inverse to K. In the case of om example, the Cremona map Mc, this means that we should compute the points MCK, Mc~lK, M2K, MC~2K, etc. Then, by the construction of Wu and W8, these points must also Ik. on both Wu and W8, since A. belonged to both Wu and WB\ Therefore the curves Wu and W8 must intersect each other over and over again. In fact, they must intersect infinitely often, and if the map is differentiable as we have been assum ing, all the angles of intersection must be nonzero. Thus the existence of a single homoclinic point implies the existence of an mfaite set of homoclinic points. And in order to achieve this irsinite set, Wu and W8 must oscillate around each other infinitely often. Figure 9 shows this intersection and oscillation for the map Afc, again with X = 3. Two properties are immediately appar ent. First, the spacing between successive homoclinic points becomes finer and finer as one approaches the hyperbolic fixed point either along Wu or W8. This is to be expected because the behavw of the map is governed by its linear part near the fixed point, and then (34) comes into play. Second, the ampli tude rf oscillation increases as one approaches the fixed point. This ©ocurs because, since the map Mc is area preserving, the areas fc*mder' successive oscillations must all be the same. In order to preserve area in the face of decreased spacing, the ampfitM.de must increase. PiK&^ties OF Area-Preserving Maps The wet effect of these two properties is that near the hyper bolic iked point the oscillations of Wu about W8 must intersect the oscillations of W8 about Wu to produce even more homo clinic points. The result is that the hyperbolic fixed point is actually the corner of an ever denser 'cloud' of homoclinic points. This property is illustrated for Mc in Figure 10 . PiK&^ties OF Area-Preserving Maps A moment's reflection on the reader's part now will show that because of the generality of our arguments, the existence of this cloud of homoclinic points is not peculiar to just the mapping Mc. It will in fact occur for any area-preserving map which possesses a homoclinic point. Homoclinic Points and Insolubility What does all this have to do with the Stormer problem? In the ne&t section we will show numerically that the mapping M Tor the Stormer problem has a homoclinic point. In this sec tion we will show that the existence of a homoclinic point means that the Stormer problem is insoluble.* Consider any dynamical system having n degrees of free dom, canonical coordinates qu • • , qn and pt, • --^(col lectively denoted by q and /?), and a time development gener ated by a time independent Hamiltonian H(p,q). Such a system will always possess 'In independent constants of motion Clip, q, t), •••, Cznip. <7, /) which in general depend on the ca nonical variables />, q and the time /. By a constant of motion we mean a function which satisfies Homoclinic Points and Insolubility along every trajectory. Indeed, given any point (q,p, t) in state space, we can always trace back the unique trajectory through this point to a fixed reference time t° and then record the In numbers/?0, #°. These quantities, which we write as/?°(/?, q. r), q°(p, q, t), are obviously constant along a trajectory by con struction. Hence they, or any 2n functionally independent functions of them, provide 2n constants of motion. By the very generality of the argument that we have just given, the existence of 2n constants of motion places very little restriction on the trajectories generated by H. Indeed it seems quite possible and is in fact assumed in statistical mechanics or ergodic theory that for some Hamiltonians there may be some trajectories which, when traced forward and backward in time, wander arbitrarily near any point in phase space. Homoclinic Points and Insolubility One way to preclude such a complicated behavior is to demonstrate the existence of what we shall call integrals of motion. For a given Hamiltonian H(p, q) we define an integral of motion I(p, q) to be a single-valued analytic time indepen dent function on phase space which, like a constant of motion, also satisfies dl/dt = 0 along every trajectory generated by H. By our hypotheses about the nature of/, equations of the form /(/>, q) = const describe a set of disjoint hypersurfaces in phase space. And if H does have / as an integral of motion, each trajectory is confined to one of these hypersurfaces. Thus complicated 'wanderings' which would take a trajectory from one hypersurface to another are ruled out. For this reason, the integrals that we have defined are sometimes called isolating integrals [Contopoulos, 1963] . Homoclinic Points and Insolubility The motion on a given hypersurface may still be very com plicated. However, suppose that there exist further indepen dent integrals /2, /3, etc. in addition to / = /t. Then trajectories must lie on intersections of families of hypersurfaces, and the 0.05 H y o - Fig. 10 . A continuation of Figure 9 near the origin showing the formation of a grid of intersecting lines. The spacing of the grid becomes finer and finer as it approaches the hyperbolic fixed point. Each grid intersection is a homoclinic point. The result of ail these intersections is an ever denser cloud of homoclinic points which has the hyperbolic fixed point as a limit point. Homoclinic Points and Insolubility • AJfc_»)«u__iJM.-__^'!5lfcF':»: f * * , 2334 Dragt and Finn: Insolubility of Orbits in a Dipole Field i motion is consequently further restricted. In general, the more integrals a system has, the simpler it is to describe its motion. And the more integrals we know for a given system, the more we know about its motion. At this point the ingenious reader may object that every Hamiltonian system with 2n degrees of freedom will automati cally possess 2n -1 integrals. One merely takes the In con stants of motion Ct(p, q, t) and functionally eliminates the time from among them. The result of this elimination will beln -1 functions of just the variables />, q, and these functions will remain constant along trajectories. Hence why not call them integrals? Homoclinic Points and Insolubility Some such argument is rpade more or less explicitly in almost every advanced mechanics text. However, all that this argument proves is the existence of what we might call local integrals of motion. There is no guarantee that these local integrals extend to global single-valued analytic functions. In fact, if the motion happens to exhibit various features of ergodicity, such an extension must be impossible. Homoclinic Points and Insolubility Let us return to the Stormer problem. It obviously possesses Pi, and H as integrals. The use of p^ enabled us to reduce the three-dimensional problem to one with two dimensions, and the use of H enabled us to group together trajectories accord ing to the value of W02. We will now show that if the mapping MJias a homoclinic point, tnenjhere are no further global integraTsj^^ This means^that there is no further singlevalued analytic function 7(p, q) on phase space which satisfies the Liouville equation Homoclinic Points and Insolubility where the brackets denote the Poisson bracket operation. ln practical calculations we begin with differential equations whose terms are analytic and thus are naturally led to work with analytic functions, or uniformly convergent (and hence again analytic) sums of analytic functions. The nonexistence of an analytic integral means that the trapped Stormer problem cannot be handled in this manner and therefore is entitled to be called 'insoluble.' This does not mean that one cannot compute Stormer trajectories. Indeed, numerical integration is always possible unless the trajectory goes to the origin. Nor does it mean that further progress cannot be made on the Stormer problem. But it does mean that further progress will be quite different from what one normally thinks about when one speaks of 'solving' a classical mechanics problem J^may in factjvenbe the case that Stdrmertrajectories are^ergodicin c e f t a i n r e g i o j i s ' o r p K *ê are ready to present an argument as to why the existence of a homoclinic point rules out the existence of any global analytic integral beyond p$ and H itself. Suppose / is a further integral. Let p be a point in the p, p plane. We have seen that such a point, for a fixed value of WV, defines a unique tra jectory in phase space. Let 7(p) denote the value of the integral along this trajectory. Now apply powers of .Af to p. Since M is generated by following trajectories and / is constant along a given trajectory, we must have hence must be continuous. Suppose p is some point on the unstable manifold Wu. Then we again get l(Mnp) = 7(p) n = 0, ±1, ±2, Homoclinic Points and Insolubility We see that / is invariant under the action of M. Suppose p is some point on the stable manifold W8 belong ing to a hyperbolic fixed point h. Then in view of (35) and (41) we must have 7 ( p ) = 7 ( h ) p c W 8 ( 4 2 ) Here we have used the fact that 7 is assumed analytic and 7(p) = 7(h) p c Wu (43) by the same argument. It follows that 7 has one and the same value, namely, 7(h), everywhere on the curves Wu and W8. In particular, the directional derivative of 7 must vanish along both the curves Wu and W8. Next suppose that Wu and W8 intersect at a homoclinic point K located at k. Then we must have Homoclinic Points and Insolubility since k belongs to both Wu and W8. But even more can be said: We know that the directional derivative of 7 is zero along both Wu and W8. However, at the point k the tangents to Wu and W8 are linearly independent vectors, since by definition 1VU and W8 intersect there at an angle. Because the directional derivative of 7 now is zero along two linearly independent directions, we conclude that the gradient of 7 must vanish at any homoclinic point: Homoclinic Points and Insolubility where k is any homoclinic point. The same argument applies to the hyperbolic point h, where W8 and Wu also meet at an angle. We are almost done. We saw earlier that the fixed point h was the corner of an ever denser cloud of homoclinic points arranged in such a way that h is an accumulation point of the cloud along several different paths. We now also know that V7 = 0 at each of these points. Finally, we have assumed that 7 and hence V7 are analytic functions. It follows from the wellknown uniqueness theorem for analytic functions [Titchmarsh, 1939] by a simple extension to two variables that V/ must vanish identically at every point in the physical region. There fore 7 must have the same value, namely, 7(h), everywhere. That is, the integral is just a constant function everywhere and is therefore useless! Thus we conclude that the existence of a homoclinic point precludes the existence of any global analytic integral except for useless constant functions [Moser, 1973] . Homoclinic Points and Insolubility We close this section with a remark about soluble classical mechanics problems such as appear in textbooks. How do they fit into our discussion? What is special about any mapping M that they might generate? Briefly, the answer to these questions is that soluble problems never produce a homoclinic point. Their fixed points are either elliptic or parabolic, so that there are no stable and unstable manifolds, or there are a few hyper bolic fixed points whose stable and unstable manifolds either never meet or, if they do meet, join smoothly without intersect ing. Thus no homoclinic points are ever formed. Figure 11 shows a periodic orbit for the Stormer problem in the case WV = 0.01. It corresponds to a fixed point h of M given by ph = 1.11494632, ph = 0. Numerical calculation shows that this point is hyperbolic with an eigenvalue given by X = 2.49 and that the eigenvectors vx and v2 of Lh are \t -.(1, 53) and \2 = (1, -53). The location of h and the arrangement of *i, v2, Wu, and W8 in this case are shown schematically in Figure 12 . Here, unlike the case in Figure 8 , nopieces of Wu or W8 go off to infinity. Instead, the piece of Wu extending from Ii into the upper half plane meets on thep"axis with that piece of W8 which approaches h from the lower half plane. The other piece of Wu also meets the other piece of W8 on the p axis with a slightly larger value ofp. In anticipation of future results we Fi^. M. A periodic Stormer orbit corresponding to a hyperbolic fixed pmnt of M. The orbit is symmetric about the p axis and retraces itselL have radicated that both intersections of Wu and W8 occur with _\ nonzero angle. That is, we have depicted two homo clinic joints. Figure 13 shows the results of computing Afn(h ± 5v1>2) foi n = -Iii-9, • • .,+10and$ = 1. X 10"M.2 X IO"6, ' • . 2.6 X 10'"*. Since b is very small, the points that we obtain in this manjinssf must lie very near Wu and W8. It is evident from the figure^hat the respective branches of Wu and W8 either meet or interact near the points (0.933, 0) and (0.945, 0), but we cannon ihe immediately sure what happens; i.e., the angles of intersection may,be zero. They are, at any rate, rather small. Numerical Evidence for Homoclinic Points By a technique to be published elsewhere [Finn, 1974] we have Ibsen abk to obtain the first few terms of a power series f(PiPs&f$2) which satisfies term by term the equation [ 7 , 7 7 ] = 0 ( 4 6 ) Numerical Evidence for Homoclinic Points Here Jf is the Hamiltonian given by (18). In addition we have been _Me to give an algorithm for computing arbitrarily many terms in the series. Our algorithm can be applied to any Ham&onian of the form (18) providing the leading piece 772 has the-structure given by (19). In actual practice the algebraic expressions involved in the calculation soon become too length .for human manipulation. For that reason we have progr_i^fned our algorithm into a digital computer and let it do the Algebra. Even the answer itself is rather complicated. For the Sttfrmer problem we find, by using the same notation as in p^, that the first few terms are given by Numerical Evidence for Homoclinic Points (47c) In gesHsxr^ it is best not to write down the answer on paper. Rather^ mne should transfer it directly from one computer program to another, since because of length, only computers can msx&e use of the answer anyhow! Threse things should be said about the power series 7. First, examination of the series shows that it contains all the terms which mme would obtain by expanding the usual adiabatic magrcte moment invariant in a power series. It also contains additkimal terms which correspond to as yet unknown (and perhaps forever unknown due to algebraic complexity) higherorder corrections to the magnetic moment invariant. Second, the series cannot be convergent if M has a homoclinic point. For if the series did converge, it would provide an additional integral of motion in contradiction to the results of the last section. Third, if we truncate the series 7 at an appropriate level, we obtain a quantity F, which is nearly constant along trajectories since 7 formally satisfies (46). This means that we may use the truncated integral F as a sort of 'magnifying glass' to examine the behavior of trajectories in great detail. The idea is that if, for example, we look at a set of points Mnp, we may not be able to detect small homoclinic oscillations because we need a rather coarse scale in order to just plot the points. However, if we instead study F(MnpX then most of the varia tion in F may be due to homoclinic oscillations. Numerical Evidence for Homoclinic Points To display how constant F is in practice, Figure 14 presents a set of points p satisfying Numerical Evidence for Homoclinic Points when F is the polynomial obtained by truncating the series 7 beyond terms of sixth order. We have replotted the points of Figure 13 to show that the outer branches of Wu and W8 lie remarkably close to the curve (48). We next select a point g given by g = (1.115, 0). It lies just slightly to the right of h. Consequently, if we compute the points Mng for n = 0, ± 1., ±2, • --, we should get points very near the outer branches of Wu and W8. We have carried out this calculation for n between -80 and +80. Figure 15 shows the results: the points are indistinguishable in behavior from those in the outer portion of Figure 13 and appear to lie on a smooth curve. Numerical Evidence for Homoclinic Points Suppose now that the outer branches of Wu and W8 intersect at a finite angle near (0.933, 0) and then go into oscillation about each other. This oscillation, if it exists, should be re flected in the behavior of the points Mng, since they lie near and k£" ""** ""' ~i"a'"'"»" =«h Ok,,,,,,™ homoclinic oscillations. Th'ile " ^ ,k,, P'aed '""" of^oforalltL^l . ■ 8 ) m u S t h e * c°m p o n e n t of Figure . S Th ■ P y'ng '" thC ri^ht half Pla™ <P*D LfTZ 3tter m°de of Pr«entation is particular y of ASi ;rd; hoart hpoi nt s near t he hype*°, i
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