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Topology of Energy Surfaces and Existence of Transversal Poincaré Sections

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Topology of Energy Surfaces and Existence of Transversal Poincaré Sections Topology of energy surfaces and existence of transversal Poincar´esections Alexey Bolsinova Holger R. Dullinb Andreas Wittekb a) Department of Mechanics and Mathematics Moscow State University Moscow 119899, Russia b) Institut f¨ur Theoretische Physik Universit¨at Bremen Postfach 330440 28344 Bremen, Germany Email: [email protected] February 1996 Abstract Two questions on the topology of compact energy surfaces of natural two degrees of freedom Hamiltonian systems in a magnetic field are discussed. We show that the topology of this 3-manifold (if it is not a unit tangent bundle) is uniquely determined by the Euler characteristic of the accessible region in arXiv:chao-dyn/9602023v1 29 Feb 1996 configuration space. In this class of 3-manifolds for most cases there does not exist a transverse and complete Poincar´esection. We show that there are topological obstacles for its existence such that only in the cases of S1 × S2 and T 3 such a Poincar´esection can exist. 1 Introduction The question of the topology of the energy surface of Hamiltonian systems was al- ready treated in the 20’s by Birkhoff and Hotelling [8, 9]. Birkhoff proposed the “streamline analogy” [3], i.e. the idea that the flow of a Hamiltonian system on the 3-manifold could be viewed as the streamlines of an incompressible fluid evolving in this manifold. Extending the work of Poincar´e[14] he noted that it might be difficult 1 to find a transverse Poincar´esection which is complete (i.e. for which every stream- line starting from the surface of section returns to it) [1]. Hotelling classified some of the topologies of energy surfaces with two degrees of freedom. 1970 Smale initiated the study of “Topology and Mechanics” [16] from the modern point of view. This work had a great influence and stimulated a lot of research especially in the Russian school of mathematics, see e.g. [7, 11, 13, 4, 18, 10]. We want to take the present knowledge about the topology of energy surfaces of natural Hamiltonian systems and return to the question of Birkhoff about the exis- tence of transverse and complete Poincar´esurfaces of section. The list of topologies of natural Hamiltonian systems is in principle known, but here we collect the results we need and give a proof using Heegard splittings which explicitly constructs an em- bedding of the split halves of our “manifold of streamlines” into R3. With the help of the computer it is possible to create a realistic picture of Birkhoffs “streamline analogy” using our result. In the second part the list of topologies of energy sur- faces is compared to the list of manifolds that can have a complete and transverse Poincar´esection, i.e. which admit the structure of a bundle over S1 with a Riemann surface as a fiber. In [6] we already noted that there can be topological obstacles for the existence of a transverse and complete Poincar´esection. We now show that in the class of all energy surfaces of natural Hamiltonian systems (possibly with a magnetic field) there can only exist a transverse and complete Poincar´esection if the 3-manifold is a direct product of S2 or T 2 with S1. 2 Topology of Energy Surfaces Consider a natural time independent Hamiltonian system with two degrees of freedom in a magnetic field. The smooth and orientable two dimensional configuration space is denoted by Q. The system is described by the Lagrangian on the tangent bundle T Q given by 1 L(q, q˙)= hq,˙ T (q)q ˙i − V (q)+ hA(q), q˙i, (1) 2 with a positive definite matrix T (q), potential V (q) and vector potential A(q). Since det T =6 0 the momenta are p = ∂L/∂q˙ and the Legendre transformation to T ∗Q gives the Hamiltonian 1 H(q,p)= h(p − A(q)), T −1(q)(p − A(q)i + V (q). (2) 2 2 If Q is compact it is a Riemann surface Rg whose genus we denote by g, otherwise 2 1 1 Q is the Euclidean plane R or a cylinder R × S . The accessible region Qh in Q for fixed energy H = h is the set of points in Q for which the potential energy does not exceed the total energy Qh = {q ∈ Q | V (q) ≤ h}, (3) which we assume to be compact. Each connected component of Qh can be treated separately. The ovals of zero velocity withq ˙ = 0 or equivalently V (q) = h are the 2 boundaries of Qh, if any. The number of ovals of zero velocity, i.e. the number of disjoint components of ∂Qh is denoted by d. By abuse of language we denote the parts of Q which are excluded from Qh by the ovals of zero velocity as “holes” in Q. The energy surface ∗ Eh = {(q,p) ∈ T Q | H(q,p)= h} (4) is compact because Qh is assumed to be compact. By this assumption we have d> 0 for Q = R2 and d > 1 for Q = R × S1. In the following we will include the cases of non-compact Q into the case of Q ≃ S2 because a disc with d − 1 holes (not counting the outer boundary of the disc) is homeomorphic to a sphere with d holes, similarly for a cylinder with d − 2 holes. Therefore the topology of Eh only depends on the genus g of Q and on the number of ovals of zero velocity d of Qh. Note that Qh is the projection of Eh onto Q. 2 The case of d = 0, i.e. the motion on a compact Riemann surface Q = Rg (with sufficiently high energy h>V (q) everywhere) almost by definition (4) has an energy 2 surface homeomorphic to the unit tangent bundle of Rg. Here we want to classify all the other cases with d> 0. Proposition 1 The topology of the energy surface Eh of a two degree of freedom Hamiltonian system is determined by the Euler characteristic χ of the accessible region of configuration space Qh if there is at least one oval of zero velocity. Our proof is elementary and constructive: We embed Q in R3 and attach ellipses of possible velocity to every point of Qh. Cutting these velocity ellipses we obtain a Heegard splitting of Eh from which the topology of Eh is determined. Since Q is an orientable Riemann surface it can be embedded in R3: Q ≃{r ∈ R3 | F (r)=0}. (5) In the Lagrangian (1) we now choose r as global coordinates with the additional constraint F (r) = 0. The energy function E˜(q, q˙) on T Q is given by 1 E˜(q, q˙)= hq,˙ T (q)q ˙i + V (q), (6) 2 and similarly 1 E(r, r˙)= hr,˙ T˜(r)r ˙i + V˜ (r), hF , r˙i =0, (7) 2 r where T˜|Q = T (q) and V˜ |Q = V (q) and the tildes are omitted in the following. The reason for treating everything on T Q instead of T ∗Q is that the linear terms in the momenta in the Hamiltonian due to the vector potential A are not present if the energy is treated as a function of the velocitiesq ˙. Moreover, note that with non- vanishing A on the boundary of Qh we have zero velocityq ˙ but not zero momentum p. 3 6 With E(r, r˙) we have an embedding of Eh into Euclidean space R given by 6 Eh ≃{(r, r˙) ∈ R | E(r, r˙)= h, F (r)=0, hFr, r˙i =0}. (8) Following Birkhoff, Hotelling and Smale [2, 8, 9, 16] the energy surface is con- structed by attaching circles in velocity space to every point in the (accessible) con- 1 figuration space Qh. This gives a fiber bundle with base Qh and fiber S where the fiber is contracted to a point on ∂Qh. In our embedding this means to take any point 3 r on Q ⊂ R and to calculate the remaining kinetic energy h − V (r). Outside Qh it is negative, on the boundary it is zero and inside of Qh it is positive. In the latter case the possible velocities are given by hr,˙ T r˙i = 2(h − V (r)). The constraint ensures thatr ˙ is in the tangent plane of F (r)=0at r. Therefore the possible velocities are located on an ellipse in the tangent plane. In order to cut the velocity ellipses at every point we need a device to fix a zero position on this S1, i.e. we want to construct a global section for the fiber bundle. This global section can be constructed with the help of a nowhere vanishing vector field ξ on Qh. On a Riemann surface Q of genus g =6 1 there does not exist a vector field ξ without equilibrium points. If, however, there are holes (or punctures) in the Riemann surface we can construct ξ on it, such that the restriction to Qh is without singularities, essentially by moving the singularities into the hole(s). Note that at this point the assumption d > 0 is necessary (except for the case of Q = T 2). Let 3 ξ(r) be specified in the embedding in R such that hξ(r), Fri = 0. Denote by N(r) the normal vector of the surface F (r) = 0. Using ξ(r) every nonzero velocity ellipse can be cut into two halves specified by hN(r),ξ(r) × r˙i ≥ 0 and hN(r),ξ(r) × r˙i ≤ 0, the two halves joining at the place where ξ andr ˙ are (anti)-parallel.
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