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SYMPLECTIC MAPS Applications SYMPLECTIC MAPS the natural generalization of the Hénon map is the First used mathematically by Hermann Weyl, the quadratic symplectic map whose normal form has been term symplectic arises from a Greek word that means given by Moser (1994). “twining or plaiting together.” This is apt, as symplectic systems always involve a pair of n-dimensional variables, the configuration q, and momentum p, which Applications are intertwined by the symplectic two form Symplectic maps arise from Hamiltonian dynamics, because these preserve the loop action. Thus, for ω = dp ∧ dq. (1) example, the time t map of any Hamiltonian flow is symplectic, as is a Poincaré return map defined on a This antisymmetric, bilinear form acts on a pair cross section. It is often easier to study the Poincaré of tangent vectors and computes the sum of the map instead of the flow, because the dimension is areas of the parallelograms formed by projecting the reduced. Even though explicit construction of the map vectors onto the planes defined by each canonical pair is typically impossible, approximation methods often (qi,pi), i = 1,...,n, giving suffice. For example, the time T map of a periodically n forced system H(q,p,t)= H(q,p,t+ T), such as a ω(v,w) = (v w − v w ). pi qi qi pi pendulum with an oscillating support, is symplectic i=1 (See Hamiltonian systems). An extreme example is H = 1 p2 − k (q)δ(t)¯ δ¯ A diffeomorphism f : X → X ona2n-dimensional 2 cos , where is the periodic Dirac manifold X with coordinates z = (q, p) is symplectic delta function; the corresponding map is the standard if it preserves the symplectic form, that is, if f ∗ω = ω map. (Arnold, 1989; McDuff & Salamon, 1995). If we As Birkhoff showed, an ideal billiard (a free particle write z = (q,p) = f(q,p), the symplectic condition moving inside a rigid, convex table) is naturally written becomes as a symplectic map. The canonical coordinates are the position and the tangential momentum at a collision point. Symplectic maps also arise naturally in systems t 0 I Df JDf = J, where J = . (2) −I 0 where the forces are localized in time or space. For example, a circular particle accelerator or storage ring has a sequence of accelerating and focusing elements Df = ∂f /∂z f J Here ij i j is the Jacobian matrix of , is that can be modeled by a composition of symplectic I n × n the Poisson matrix, and is the identity matrix. maps, providing the damping effects of radiation can Equivalently, Stokes’ theorem can be used to show that be neglected (Forest, 1998). A[γ ]= p q f the loop action, γ d , is preserved by for Area-preserving maps also arise in the study of the γ X f any contractible loop on .If preserves the loop motion of Lagrangian tracers in incompressible fluids action for all loops, even those that are not contractible, or of particles tightly gyrating around magnetic field then it is exact symplectic. lines. In particular, when one component of the field is n = When 1, the symplectic condition is equivalent particularly strong, such as in the plasma device called (Df ) = to det 1, so that the map is area- and a tokamak, the transverse dynamics reduces to an area- orientation-preserving. Examples include the much preserving map. studied standard map and the area-preserving Hénon 2 Autonomous canonical transformations are sym- quadratic map f(q,p)= (p + a − q , − q) (Meiss, F(q,q) n> plectic maps. For example, if is a generating 1992). When 1, the symplectic condition implies function for a canonical transformation, then it gener- volume and orientation preservation, but as we will see, ates a symplectic map. In particular, the Froeschlé map is stronger than this. A generalization of the standard (3) is generated by F(q,q) = 1 (q − q)2 − V(q). map to higher dimensions is the map 2 An algorithm that respects the symplectic nature q = q + p −∇V(q), of Hamiltonian dynamics is called a symplectic integrator. A first-order symplectic algorithm with p = p −∇V(q), (3) time step t for the Hamiltonian H(q,p) is F(q,p) = qp + tH (q, p) where q ∈ Tn is an angle, p ∈ Rn is its conjugate generated by where F = q p + p q momentum, and V(q) is a periodic potential. This d d d , giving the map map is exact symplectic for any V . Beginning in 1972, Claude Froeschlé studied the case n = 2 and ∂H ∂H q = q + t (q, p ), p = p − t (q, p ). (4) V(q)= a cos q1 + b cos q2 + c cos(q1 + q2). Similarly, ∂p ∂q 1 SYMPLECTIC MAPS Note that the map is implicit since H is evaluated Symplectic Geometry at p. However, for the case that H = K(p)+ V(q) Every symplectic map is volume- and orientation- this becomes a leap-frog Euler scheme, an example preserving, but the group Symp(X) of symplectic of a “splitting” method. Symplectic versions of many diffeomorphisms on X is significantly smaller than that standard algorithms—such as Runge–Kutta—can be of the volume-preserving ones. This was first shown in obtained (Marsden et al., 1996). While there is still 1985 by Gromov in his celebrated “nonsqueezing” (or some controversy on the utility of symplectic methods symplectic camel) theorem. Let B(r) be the closed ball versus methods that, for example, conserve energy and n of radius r in R2 and C (R) ={(q, p) : q2 + p2 ≤ R2} other invariants or have variable time-stepping, they are 1 1 1 be a cylinder of radius R whose circular cross section is superior for stability properties since they respect the a symplectic plane. Since the volume of C is infinite, spectral properties of the symplectic group. 1 it is easy to construct a volume-preserving map that takes B(r) into C1(R) regardless of their radii. What Gromov showed is that it is impossible to do this symplectically whenever r>R. This is one example The Symplectic Group of a symplectic capacity, and is leading to a theory of symplectic topology (McDuff & Salamon, 1995). Another focus of this theory is to characterize The stability of an orbit {...zt ,zt + 1, ...} where the number of fixed points of a symplectic map, zt + 1 = f(zt ) is governed by the Jacobian matrix of f evaluated along the orbit, M = t Df (zt ). that is, to generalize the classical Poincaré–Birkhoff When f is symplectic, M obeys (2), Mt JM= J . theorem for area-preserving maps on an annulus. The set of all such 2n × 2n matrices form the Arnold conjectured in the 1960s that any Hamiltonian symplectic group Sp(2n). This group is an n(2n + 1)- diffeomorphism on a compact manifold X must have dimensional Lie group, whose Lie algebra is the at least as many fixed points as a function on X must set of Hamiltonian matrices—matrices of the form have critical points. A Hamiltonian map is a symplectic JS where S is symmetric. Thus, every near-identity map that can be written as a composition of maps of the symplectic matrix can be obtained as the exponential form (4). Conley and Zender proved this in 1985 for the of a Hamiltonian matrix and corresponds to the time t- case that X is the 2n-torus: f must have at least 2n + 1 2n map of a linear Hamiltonian flow. There are symplectic fixed points (at least 2 if they are all nondegenerate) matrices, however, that are not the exponentials of (Golé, 2001). Hamiltonian matrices; for example, − I.As a manifold, the symplectic group has a single nontrivial loop (its fundamental group is the integers). The winding Dynamics number of a loop in the symplectic group is called In general, the dynamics of a symplectic map consists the Maslov index (McDuff & Salamon, 1995); it is of a complicated mixture of regular and chaotic especially important for semi-classical quantization. motion (Meiss, 1992). Numerical studies indicate that If M is a symplectic matrix and λ is an eigenvalue the chaotic orbits have positive Lyapunov exponents of M with multiplicity k, then so is λ − 1. Moreover and fill sets of positive measure that are fractal in det(M) = 1, so M is volume and orientation preserving. nature. Regular orbits include periodic and quasi- A consequence of this spectral theorem is that orbits of a periodic orbits. The latter densely cover invariant tori symplectic map cannot be asymptotically stable. There whose dimensions range from 1 to n. Near elliptic are four basic stability types for symplectic maps: an periodic orbits, the phase space is foliated by a eigenvalue pair (λ, λ − 1) is positive-measure cantor set of n-dimensional invariant tori. There are chaotic regions in the resonant gaps • λ hyperbolic,if is real and larger than one; between the tori, but the chaos becomes exponentially • λ hyperbolic with reflection,if is real and less than slow and exponentially small close to the periodic minus one; orbit. Some of these observations, but not all, can be • λ = 2πiω elliptic,if e has magnitude one; proved. • part of a Krein quartet if λ is complex and has The simplest case is that of an integrable symplectic magnitude different from one, for then there is a map, which can be written in Birkhoff normal form: quartet of related eigenvalues (λ, λ − 1, λ,¯ λ¯ − 1). f(θ,J)= (θ +∇S(J),J). Here (θ, J ) are angle- Thus, a periodic orbit can be linearly stable only when action coordinates (each n-dimensional) and =∇S is all of its eigenvalue pairs are elliptic.
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