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SN~ (~6) lff 2It 3k_~ , 5 ",,x_J {b)

SN~ (~6) lff 2It 3k_~ , 5 ",,x_J {b)

Caustics in Time

Caustics in Time Reversible Hamiltonian Systems James Montaldi Abstract We consider the projection to configuration space of invariant tori in a time reversible Hamiltonian system at a point of zero momentum. At such points the projection has rank zero and the resulting caustic has a corner. We use caustic equivalence of Lagrangian mappings to find a normal form for such a corner in 3 degrees of freedom. 1 Introduction Invariant torl arise in many Itamiltonian systems. For example, the closure of any bounded trajectory in a completely integrable system is a torus, and if a completely integrable system is slightly perturbed then by KAM theory many of these invariant tori persist, [2]. Invariant tori also abound near stable equilibrium points and stable periodic orbits. There is considerable interest in the geometry of the projections of invaria~lt tori into configuration space, particularly if the system is a classical Hamiltonian system (i.e., of the form 'kinetic + potential'), see for example, [1], [7] and [10]. This paper carries further some work of J.B. Delos [7] in which he explains why pro- jections of invariant tori in classical Hamiltonian systems with 2 degrees of freedom can have corners. At first sight, from a singularity theoretic point of view, one would expect the projection of an invariaalt 2-toms on to the configuration plane to have only fold and cusp singularities. However, Delos shows that in classical ttamiltonian systems if the toms contains a point (q,p) = (q, 0) (where q and p are conjugate position and momentum) then near such a point the projection (q,p) ~ q of the toms has a 'folded handkerchief' singu- larity, that is, in suitable local coordinates (z, y) about this point, the projection takes the form (z, y) ~ (x 2, y2). The singularities of the projection of tori at points where p y~ 0 are of the expected type: folds or cusps. See Figure 1 for two typical trajectories in 2 degrees of freedom, one with these corners and one without. Further pictures of such projections can be found in the references cited above. It turns out that the crucial property of a Hamiltonian system which causes comers to appear generically is time-reversibility; it is not strictly necessary for the system to be in the classical 'kinetic + potential' form, though because kinetic energy is even in momentum any classical system is time reversible. (In fact, I do not know of an interesting Hamiltonian system which fulfills the requirements of this paper but is not of the classical form.)

267 In this paper I consider invariant tori, or possibly more generally, invariant Lagrangian submanifolds, in time reversible systems, but now in 3 degrees of freedom. The main result is that the analogue of corners in 3 degrees of freedom is not just the corner of a cube (as Delos conjectured) but is more subtle, see Figure 2. The technique required in 3 degrees of freedom is also more subtle. In 2 degrees of freedom, the results use the standard theory about Lagrangian singularities via T~+-equivalence of generating families, adapted to allow for the time-reversal symmetry; however in 3 degrees of freedom, the generating families have infinite 7~+-codimension, so it is necessary to weaken the equivalence relation to 'caustic equivalence', which involves J. Damon's theory of/Cv-equlvalence. Acknowledgements. This research was supported by a grant from the SERC. I would like to thank Mark Roberts for bringing the work of Delos to my attention, and also for many stimulating discussions. The computer-generated pictures were obtained using Guckenheimer and Kim's Kaos package (implemented on a Sun computer). 2 Invariant tori in reversible systems We consider time reversible systems on a phase space T*Q which is the cotangent bundle of an n-dimensional configuration space Q. We denote the natural projection by r :T*Q --* Q. The time-reversal involution ~" : T*Q -~ T*Q is given by v(q,p) = (q,-p), where q E Q and p E T~Q. We assume that the Hamiltonian H : T*Q --4 R is invariant under 7" for then the associated Hamiltonian vector field X is reversed: r,X(q,p) = -X(q,-p). This implies that if t ~-, 7(t) is an integral curve of X then so is t ~ 7- o 7(-t). Note that for p = 0, we have r.X(q,O) = -X(q,O), so X(q,0) is 'vertical' for all q E Q. It follows from these observations that if 7(0) = (q, 0) then the closure of the set {7(t) I t E R} is invariant under the involution r, in which case we say it is time reversible. We are interested in the case that the integral curve 7 is dense in a Lagrangian sub- anlfold which we will denote by L (for example, 7 is a quasiperiodic orbit whose closure is a torus of dimension n). If ]5 meets the fixed point set of r -- the zero-section of T'Q, which, abusing notation, we will denote by Q -- then ]5 is r-invariant: it is a time reversible Lagrangian submanifold. Proposition 1 If q E ]5 and L thq Q, then r acts as -I in a neighbourhood of q in L, and since r(z) = r(r(z)) we have that rlL(--z ) = rlL(Z ) for z in this neighbourhood. That is, rlL is an even map. 0 We now show that if L is a time reversible torus which is transverse to (and meets) Q then/) fl Q consists of precisely 2 n points, where n = dim Q. Proposition 2 Let L be a torus of dimension n, and r : L -~ L an involution with only isolated fixed points. If r has one fized point, then it has precisely 2 n fixed points. PROOF: This follows from the Lefschetz fixed point theorem in an appropriate form. How- ever, since in this case the argument is particularly simple and there is not a good reference, I will outline a proof from first principles. Let N be the number of fixed points of r. The involution r generates the group 72. Consider the quotient space L/72. This has N

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