Symplectic maps

Post-publication activity

Dr. Christophe Golé, Mathematics and Statistics, Smith College

A symplectic map is a diffeomorphism that preserves a symplectic structure.

The simplest example of symplectic map is a map \(F:{\mathbb R}^2 \to {\mathbb R}^2\) which preserves the area and orientation, i.e. such that for all points \( z\in {\mathbb R}^2\) we have \(\det D_zF = 1\ ,\) where \(D_zF\) is the differential (or Jacobian matrix) of \(F\ .\) In terms of differential forms, this can be expressed as \(F^*(dx\wedge dy) = dx \wedge dy.\)

More generally, if \(M\) and \(N\) are manifolds of dimension \(2n\) and \(\omega_M, \omega_N\) are symplectic forms (non-degenerate, closed, differentiable 2-forms) on \(M\) and \(N\ ,\) then a diffeomorphism \(F: M\to N\) is a symplectic map if \[\tag{1} F^*\omega_N = \omega_M. \]

By a theorem of Darboux, for each point \( z\in M\) one can always find local coordinates such that identity (1) translates to: \[(D_zF^T) J (D_zF) = J,\] where \( J = \begin{pmatrix} 0_n & I_n \\ -I_n & 0_n \end{pmatrix} \)

Symplectic maps arise naturally in physical systems. In particular, they are closely related to Hamiltonian systems. Other names for symplectic maps are canonical transformations and symplectomorphisms. Symplectic maps are central to the theories of Kolmogorov-Arnold-Moser (KAM), Aubry-Mather and symplectic topology.

Historical Motivation: the 2-Dimensional Case

Poincaré Section

When studying periodic orbits in Celestial Mechanics, Poincaré introduced the notion of first return map on a surface, also called a Poincaré section, transverse to a periodic orbit at a point \(z^*\ .\) The orbits of points of the section close to \(z^*\) return to it, defining a map from a punctured neighborhood of \(z^*\) of the section to itself. Because mechanical systems are Hamiltonian, when the system has two degrees of freedom, the resulting map preserves area (Meiss (1992)). Under certain conditions on the orbit of \(z^*\) (e.g. that it be elliptic), this map gives rise to a simple model of area preserving maps on the closed cylinder (or annulus) \(\mathbb S^1 \times [-1, 1]\) with the boundary twist condition: points on the boundaries are rotated in opposite directions.

Poincaré-Birkhoff Theorem and Symplectic Topology

A year before his death in 1912, Poincaré proposed an incomplete proof of a theorem stating that area preserving maps of the annulus with the boundary twist condition have at least two fixed points (Birkhoff gave another proof the following year). This result contrasts with Lefschetz' fixed point theory: the topology of the annulus does not guarantee the existence of fixed points for homeomorphisms of the cylinder, as the example of a rotation shows. A proof of the Poincaré-Birkhoff Theorem in the simple case of a positive twist map (on the cylinder, homeomorphic to the annulus), illustrates the connection with variational calculus and Morse Theory characteristic of the more recent field of symplectic topology.

Figure 1: The curve \(C\) of points of the cylinder that are displaced vertically by \(F\ .\) Fixed points of \(F\) are the intersection of \(C\) and \(F(C)\ .\)

An area preserving positive twist map of the cylinder \(\mathbb S^1\times[-1,1]\) is an area preserving map \(F:(q,p)\mapsto (Q, P)\) such that \(\frac {\partial Q}{\partial p}>0\ :\) as one moves up, the image of the point veers right. Assuming that the points on the lower boundary and upper boundaries move in opposite directions, there must be exactly one point \(z(q_0)\) on each vertical segment \(\{q = q_0\}\) that moves only vertically. The curve \(C\) parametrized by \(q_0\to z(q_0)\) is homeomorphic to a circle. Because area is preserved, \(C\cap F(C)\) must contain at least two points, which must be fixed points of \(F\ .\) These points also are the critical points of the "generating" function \(S(q) = \int_{q_0}^q PdQ - pdq\) on \(C\ ,\) which computes the area between a segment of \(C\) and its image by \(F\ .\) Morse theory and its generalizations imply that a differentiable function on the circle has two critical points, insuring the existence of at least two fixed points for \(F\ .\)

One can prove the general case of the Poincaré-Birkhoff theorem by decomposing the map into positive and negative twist maps.

This theorem gave rise to the famous conjecture of V. I. Arnold: a (Hamiltonian) symplectic map on a compact, closed symplectic manifold has as many fixed points as a smooth function has critical points. This conjecture was proved in the special case of the standard torus by C. Conley and E. Zehnder in 1983. This, together with M. Gromov's seminal paper (1985) on \(J\)-holomorphic curves, marked the beginning of the field of symplectic topology. A. Floer (1989) made the essential breakthrough in establishing the general case. After work by many other mathematicians, the full conjecture was finally proven by G. Liu and G. Tian (1998) and K. Fukaya and K. Ono (1996).

KAM and Aubry-Mather Theories

Area preserving twist maps led J. Moser to an instance of the famous Kolmogorov-Arnold-Moser (KAM) theory. Consider the map of the cylinder \(F_0(q,p) = (q+p,p)\ .\) This area-preserving map is also completely integrable\[F_0\] preserves each circle \(\{p = p_*\}\) on which it induces a rotation by angle \(p_*\) (measured in fraction of circumference). If \(p_*\) is rational, the circle is a union of periodic orbits. If \(p_*\) is irrational, each orbit is dense on the circle. KAM theory implies that, for an area-preserving map \(F_\epsilon\) close to \(F_0\ ,\) there are \(F_\epsilon\)-invariant circles of all ``sufficiently irrational" rotation number which fill a large proportion (in measure) of the cylinder.

The theory of Aubry-Mather shows that, when these invariant irrational circles break down, the map still has invariant Cantor sets, on which it acts as circle homeomorphisms of Denjoy type do on their recurrent set. Together with the Poincaré-Birkhoff Theorem, one can thus show the existence of orbits of all rotation numbers in the interval whose bounds are given by the rotation numbers of the map on the boundaries. These orbits have order properties and minimize an action defined via the generating function. However, between the invariant circles that do remain, chaotic motion occurs.

Linear Symplectic Theory

Linear Symplectic Forms

A linear symplectic form on a vector space is bilinear form \(\omega\) which is

skew symmetric\[\omega(w, v) = -\omega(v, w)\]
non-degenerate: if \(\omega(v, w) = 0\) for all \(w\) then \(v=0\ .\)

These conditions imply that the space is of even dimension \(2n\ .\) Given a basis, to \(\omega\) one can associate a non-degenerate \( 2n \times 2n\) skew symmetric matrix \(A\) by \(\omega(v,w) = <Av,w>\ ,\) where \(<\quad, \quad >\) is the usual dot product in the basis. By a process similar to that of Gram-Schmidt, one can always find coordinates, called Darboux coordinates, \((q_1, \ldots, q_n, p_1, \ldots, p_n)\) so that \[A = J = \begin{pmatrix} 0_n & I_n \\ -I_n & 0_n \end{pmatrix} \quad \text{which means} \quad \omega = \sum_{k=1}^n dq_k\wedge dp_k\]

Note that \(dq_k\wedge dp_k (v,w)\) is the determinant of the \(2\times 2\) matrix formed by the \(k^{th}\) and \((n+k)^{th}\) components of \(v\) and \(w\ .\) Also, the \(n^{th}\) wedge power of this symplectic form is related to the standard volume form in these coordinates: \[\tag{2} \omega^n = {(-1)^{[n/2]}}{n!} [dq_1\wedge dp_1 \wedge\ldots \wedge dq_n\wedge dp_n] \]

Symplectic Linear Maps and Matrices

A symplectic linear map \(L\) on \(\mathbb R^{2n}\) is a linear map that preserves the symplectic form: \[ \omega(Lv, Lw) = \omega(v, w)\] In Darboux coordinates, where \(\omega(v,w) = <Jv,w>\ ,\) then \(L\ ,\) seen as a matrix, satisfies: \[\tag{3} L^TJL = J. \]

We say that the matrix \(L\) is symplectic.

Properties of Symplectic Matrices

Because of Equation (), a symplectic linear map preserves volume and orientation. Hence the map has determinant 1 and it is invertible.
From Equation (3), if \(\lambda\) is an eigenvalue with multiplicity \(k\) of \(S\) then so are \(\lambda^{-1}, \overline\lambda, \overline\lambda^{-1}\ .\) This has important consequences on the stability of periodic orbits of Hamiltonian systems and symplectic maps. (see Stability of symplectic maps).
If we write \(L = \begin{pmatrix} a & b\\ c & d \end{pmatrix}\) in the \(n\times n\) blocks representation corresponding to the splitting of the coordinates as \((q,p)\) where \(q,p\in {\mathbb R}^n\ ,\) then:

\[ ab^T = ba^T, cd^T = dc^T, ad^T-bc^T = I_n \text{ and } L^{-1} = \begin{pmatrix} d^T & -b^T\\ -c^T & a^T \end{pmatrix}\]

The Symplectic Group \(Sp(2n)\)

Symplectic matrices form a (Lie) group called the Symplectic Group \(Sp(2n)\ ,\) whose Lie Algebra is the set of Hamiltonian matrices, matrices of the form \( JS\) where \(S\) is symmetric. Thus every near-identity symplectic matrix can be obtained as the exponential of a Hamiltonian matrix and corresponds to the time \(t\)-map of a linear Hamiltonian flow. There are symplectic matrices, however, that are not the exponentials of Hamiltonian matrices, for example, \(\begin{pmatrix} -1&1\\ 0& -1\end{pmatrix}\ .\)

The intersection of \(Sp(2n)\) with the orthogonal group \(O(2n)\) is a group isomorphic to the complex unitary group \(U(n)\ :\) \[ U(n) =Sp(2n)\cap O(2n)= Sp(2n) \cap GL(n,\mathbf{C}) =O(2n)\cap GL(n,\mathbf{C}). \] If \(L\) is a symplectic matrix, it can be decomposed into \(L = PO\) where \(P = (LL^T)^{1/2}\) is symmetric, positive definite and symplectic, and \(O= LP^{-1}\) is orthogonal and symplectic, and hence unitary. The path \(\alpha\to P^\alpha O\) provides a deformation-retraction of \(Sp(2n)\) onto \(U(n)\ .\) Thus \(Sp(2n)\) inherits the topology of \(U(n)\ ,\) and in particular \(\pi_1(Sp(2n)) = \mathbb Z\ .\) The winding number of a loop of symplectic matrices in \(Sp(2n)\) is called the Maslov index.

Symplectic Maps and Symplectic Manifolds

Symplectic Manifolds

A symplectic structure on a manifold \(M\) is given by a closed, non-degenerate, differentiable two-form \(\omega\ .\) This means \(d\omega =0\) and for all tangent vectors \(v\) to \(M\ ,\) there is another vector \(w\) tangent to \(M\) at the same point such that \(\omega(v,w) \neq 0.\)

The remarkable theorem of Darboux states that, around any point of a symplectic manifold, one can always find Darboux local coordinates \((q_1, \ldots, q_n, p_1, \ldots, p_n)\ ,\) where, as in the linear case, \(\omega = \sum_{k=1}^n dq_k\wedge dp_k\ .\) Thus, in stark contrast to the curvature in Riemannian geometry, there are thus no local invariants in symplectic geometry.

Examples of Symplectic Manifolds

Orientable surfaces, with \(\omega\) given by their area form.
Kähler manifolds (manifolds with a Hermitian metric with a certain integrability condition, which includes all the algebraic varieties (embedded in \(CP^n\)).
Cotangent bundles of differentiable manifolds.

The cotangent bundle \(T^*M\) of a differentiable manifold \(M\) is the vector bundle with base \(M\) which is dual to the tangent bundle \(TM\ :\) each fiber \(T^*_qM\) is the vector space of one-forms acting on the corresponding tangent space \(T_qM\ .\) In physics, \(M\) often constitutes the spatial coordinates of a system and \(T_q^*M\) corresponds to the possible momenta at position \(q\ .\) The cotangent bundle is a manifold in its own right, with coordinates \((q_1, \ldots, q_n, p_1, \ldots, p_n)\ ,\) where one chooses the coordinate function \(p_k\) so that \(p_k(\sum a_j dq_j) = a_k\) on any one-form \(\sum a_j dq_j\) (the q and p are then called conjugate coordinates). The one-form \(\alpha = pdq= \sum p_k dq_k\) on \(T^*M\ ,\) often called the Liouville form, is canonical: it has a coordinate free definition and thus does not depend on the choice of conjugate coordinates. The canonical symplectic form is then \(\omega = -d\alpha\ ,\) and conjugate coordinates are Darboux coordinates for \(\omega\ .\)

Symplectic Maps Between Manifolds

If \((M, \omega_M)\ ,\) \((N, \omega_N)\) are two symplectic manifolds of dimension \(n\ ,\) a diffeomorphism \(F: M \rightarrow N\) is symplectic if \[ F^*\omega_N = \omega_M \] where the pull-back form \(F^*\omega_N\) is defined on \(TM\) by \(F^*\omega_N(v,w) = \omega_N(DF(v), DF(w))\ .\) If the manifolds \(M\) and \(N\) are exact symplectic, i.e. if there are 1-forms \(\alpha_M, \alpha_N\) with \(d\alpha_M = \omega_M,\ d\alpha_N = \omega_N\ ,\) then \(F\) is exact symplectic if \[ F^*\alpha_N - \alpha_M = dS \] for some real valued function \(S\) on \(M\ .\) The identities \(d(F^*\alpha_N)= F^* (d\alpha_N)\) and \(d(dS) =0\) imply that an exact symplectic map is symplectic.

Properties of Symplectic Maps

At each point \(z\ ,\) the differential \(D_zF: TM \rightarrow TN\) of a symplectic map is a symplectic linear map. In local Darboux coordinates \((q_1, \ldots q_n, p_1, \ldots p_n)\) this is equivalent to:

\[ (D_zF^T) J (D_zF) = J, \] with \( J = \begin{pmatrix} 0_n & I_n \\ -I_n & 0_n \end{pmatrix} \ .\) Thus, seen as a matrix, \( D_zF\) satisfies the properties of symplectic matrices enumerated above.

In particular, symplectic maps are volume preserving diffeomorphisms.

If the symplectic form \( \omega = d\alpha\) is exact on \(M\) (e.g. \(M\) is a cotangent bundle), then, by Stokes' Theorem \(F:M\rightarrow M\) is symplectic if and only if \( \oint_{F(\gamma)} \alpha = \oint_\gamma \alpha \) for any contractible differentiable loop on \(M\ ,\) and \(F\) is exact symplectic if and only if this equality is true for any (not necessarily contractible) differentiable loop.

A map \(F : N \rightarrow N\) is symplectic if it preserves the Poisson bracket\[ \{f , g\}\circ F = \{f\circ F, g\circ F\} \] where \(f, g\) are any two differentiable real valued functions on \(N\) and in Darboux coordinates \(\{f,g\} = \sum_{i=1}^{n} \left[ \frac{\partial f}{\partial q_{i}} \frac{\partial g}{\partial p_{i}} - \frac{\partial f}{\partial p_{i}} \frac{\partial g}{\partial q_{i}} \right]. \)

If \(F,G: M\rightarrow M\) are two symplectic maps, then \(F\circ G\) is also symplectic. Since symplectomorphisms on a given manifold are invertible, they form a group, that we denote by \(Symp(M,\omega)\ .\)

Examples of Symplectic Maps

Area preserving maps between two surfaces.
The pull-back map \(f^*:T^*N \rightarrow T^*M\) induced by a diffeomorphism \(f: M \rightarrow N\ .\) In local coordinates, \(f^*\) maps the fibers of \(T^*N\) linearly to those of \(T^*M\ ,\) with the matrix \(Df^T\ .\)
Time-t maps of the flows of Hamiltonian dynamical systems (follow the link for physical examples). This type of map, historically the most important in this field, receives more attention below.

Symplectic vs. Hamiltonian diffeomorphisms

Suppose \(F: M\rightarrow M\) is symplectic, in the connected component of the identity \(Symp_0(M,\omega)\) of the group of symplectic maps. This implies that \(F = \psi_1\) for a symplectic isotopy (a differentiable curve of symplectic maps) \(t\rightarrow \psi_t, t\in [0,1]\ ,\) with \(\psi_0 = Id\ .\) Let \(X_t\) be the time dependent vector field on \(M\) defined by

\[ \frac d{dt} \psi_t = X_t\circ\psi_t. \] Reminding the reader that \(i_{X_t}\omega(.) \equiv \omega(X_t, .)\ ,\) the homotopy formula of exterior calculus: \[\tag{4} \frac d{dt} \psi_t^* \omega = \psi_t^*(i_{X_t}d\omega +d(i_{X_t}\omega)) \]

implies that \(d(i_{X_t}\omega)=0\ ,\) since \(\psi_t^*\omega = \omega\) is constant and \(d\omega=0\ .\) Thus the form \(i_{X_t}\omega\) is closed. When it is exact, i.e. \[ i_{X_t}\omega = dH_t, \] for some (time dependent) function \(H_t: M\rightarrow R\ ,\) \(F\) is called a Hamiltonian diffeomorphism with Hamiltonian function \(H_t\ ,\) and \(X_t\ ,\) often denoted \(X_{H_t}\ ,\) is the Hamiltonian vector field. Going in the opposite direction with this reasoning, and assuming \(X_t\) is a Hamiltonian vector field, \(d(i_{X_t}\omega) = d^2H_t=0\) and using the homotopy formula (4), one obtains that a Hamiltonian diffeomorphism is automatically symplectic. The set \(Ham(M,\omega)\) of Hamiltonian diffeomorphisms also forms a group, hence a subgroup of \(Symp_0(M,\omega)\ .\)

Traditionally symplectic maps (often called "canonical" in this context) have been used as changes of coordinates for Hamiltonian systems\[F\] is symplectic if and only if it preserves any Hamiltonian vector field\[F_*X_{H\circ F} = X_H\ .\]

A natural question arises: what characterizes the symplectic diffeomorphisms that are Hamiltonian? The non-compact case is less understood than the compact one. Below, we give partial answers to this question for symplectic twist maps of cotangent bundles. If the manifold \(M\) is compact

and without boundary however, one can prove that \(F\in Symp_0(M,\omega)\) is Hamiltonian if and only if there exists a symplectic isotopy \(\psi_t\in Symp_0(M,\omega)\) such that \(\psi_0 = Id, \psi_1 = F\) and \({\rm Flux}(\psi_t)\equiv\int_0^1 [i_{X_t}\omega ] dt =0\) where \([i_{X_t}\omega ]\) denotes the cohomology class of \(i_{X_t}\omega\ .\) The flux of the isotopy is a group homomorphism between the covering space of \(Symp_0(M, \omega)\) and the cohomology group \(H^1(M;R)\ .\) Since \(H^1(M;R)\) is abelian, the commutator subgroup of \(Symp_0(M,\omega)\) is contained in \(Ham(M, \omega)\ .\) From a theorem by Banyaga (see McDuff & Salamon (1998)), these two groups are in fact equal: \[[Symp_0(M,\omega), Symp_0(M,\omega)]=Ham(M, \omega).\]

Symplectic Topology

Symplectic topology studies global phenomena that distinguish symplectic maps and their group from other groups of diffeomorphisms. For more on this subject, see McDuff & Salamon (1998)

\(Ham(M, \omega)\) vs. \(Symp(M, \omega)\) vs. \(Diff(M, \omega)\ .\) If \(M\) is compact and without boundary it was proved by K. Ono in (2006) that the group \(Ham(M, \omega)\) is closed in \(Symp(M,\omega)\) with respect to the \(C^1\) topology. It is much easier to see that \(Symp (M,\omega)\) is \( C^1\) closed in the group \(Diff(M)\) of diffeomorphisms. It is even closed in \(Diff(M)\) for the \(C^0\) (or uniform) topology, a deep result of Eliashberg.

\(Symp(M, \omega)\) vs. \(Vol(M, \omega)\ .\) In dimension 2, the concepts of area preserving and symplectic maps coincide. In higher dimensions, it turns out that volume preserving maps may or may not be symplectic. A distinction is given by Gromov's famous Non-squeezing Theorem: in \( {\mathbb R}^{2n}, n>1\) one cannot embed a unit ball inside an appropriately constructed, narrow enough cylinder, whereas this is always possible with a volume preserving diffeomorphism. The proof of this theorem gave rise to the notion of "symplectic width" and "capacities".

Arnold Conjecture. If you glue two annuli along their boundaries, you get a torus. This led V. I. Arnold to conjecture, based on the Poincaré-Birkhoff Theorem, that a Hamiltonian map on the torus has 4 fixed points if all are non-degenerate. And, more generally, that a Hamiltonian diffeomorphism on a compact symplectic manifold must have at least as many fixed points as a real valued function on that manifold has critical points (sum of Betti numbers if non-degenerate, cup length in general). Conley and Zehnder (1982) proved the case of the torus. Subsequent work on this conjecture gave rise to Floer homology and the full conjecture was finally proven by Liu and Tian (1998) and by Fukaya and Ono (1999).

Symplectic Twist Maps on Cotangent Bundles

Symplectic twist maps are a natural generalization of area preserving positive (or negative) twist maps of the cylinder. Indeed, the cylinder can be seen as the cotangent bundle of the circle.

Definition

Let \(U\subset T^*M\) be a tubular neighborhood of the base \(M\subset T^*M\ ,\) and \(\pi: T^*M\rightarrow M\) be the canonical projection.

A diffeomorphism \(F: U\rightarrow U\) is called a symplectic twist map if:

Figure 2: The twist condition: a fiber \(U_{q_0}\) is mapped to a graph over the base M.

F is homotopic to the Identity
F is exact symplectic\[F^*pdq - pdq = dS\] for some \(S: U \rightarrow \mathbb R\)
(Twist Condition) For every fiber \(U_{q_0}\) of \(U\ ,\) the map \(\pi\circ F: U_{q_0} \rightarrow M\) is an embedding.

Letting \(q,p\) denote the base and fiber local variables respectively, and \( F(q,p) = (Q, P)\ ,\) the twist condition implies that \(p \rightarrow Q(q_0, p)\) is a local diffeomorphism and thus \(\frac{\partial Q}{\partial p}\) is a non-degenerate matrix. In certain cases, a non-uniform non-degeneracy is sufficient to obtain a (global) twist condition. When the universal cover of \(M\ ,\) \(\tilde M = {\mathbb R}^{n}\ ,\) as in the important case of \(M = {\mathbb T}^n\ ,\) one can make this definition more global and ask that \(U= T^*{\tilde M}\) and that \(\pi\circ \tilde F: T^*_{q_0}{\tilde M} \rightarrow {\tilde M}\) be a diffeomorphism for every \(q_0\ .\)

The correspondence:

\[\tag{5} F(q,p) = (Q, P)\Longleftrightarrow p = -\partial_1S(q, Q), \quad P = \partial_2S(q,Q) \]

is a homeomorphism between symplectic twist maps and the set of functions \(S\) (defined on the appropriate subset of \(M\times M\)) satisfying \(\partial_{1}\partial_{2}S\) is uniformly non degenerate. Equation (5) implies that, for a symplectic twist map \(F\) with generating function \(S\ ,\) the following variational principle holds:

\[ (q_0,p_0), \ldots, (q_N, p_N) \] is an orbit segment of \( F \Longleftrightarrow (q_0, \ldots , q_N) \) is a critical point for \( W(q_0, \ldots , q_N) = \sum_{k=0}^{N-1} S(q_k, q_{k+1}) \) with the correspondence given by \(p_k = -\partial_1S(q_k, q_{k+1})=\partial_2S(q_{k-1}, q_{k}).\) This correspondence is a discrete analog of the Hamiltonian-Lagrangian correspondence in continuous time systems, the twist condition may be seen as a weak form of the Legendre condition and \(W\) as an action functional on discrete paths.

Examples

The Generalized Standard Map

Let \(V(q)\) be a \(\mathbb Z^n\)-periodic function. Then the map \(F: \mathbb R^{2n}\rightarrow {\mathbb R}^{2n}\) given \[ F(q , p) = (q+p+\nabla V(q), p +\nabla V(q)) \] has generating function \(S(q,Q) = \frac 12 \| Q -q \|^2 + V(q)\ .\) By periodicity, \(F\) is the lift of a symplectic twist map on \(T^*\mathbb T^n \cong {\mathbb T}^n\times\mathbb R^n\ .\) When \(V\equiv 0\) then the map is completely integrable: it acts on each torus \(p = p_0\) as a translation with ``rotation vector" \(p_0\ .\) When \(n =1\) and \(V(q) = K\cos(q)\) one obtain the (Chirikov) Standard Map. An enormous amount has been written about this map as it contains a microcosm of much of Hamiltonian dynamics. The function \(W\) for its variational counterpart also appears in the Frenkel-Kontorova model, whose study by Aubry augured the Aubry-Mather theory.

Billiard Maps

Consider the dynamics of a ball on a convex billiard table with smooth boundary \(C(q)\ .\) Let \(\theta\) be the angle of rebound and \(p = -\cos(\theta)\ .\) Then the map \(F(q,p) = (Q, P)\) associating a point of rebound and angle to the next is a twist map on \(S^1\times (-1,1)\) with generating function \(S(q, Q) = \| C(Q) - C(q) \|\ .\) Hence \(W\) is the length of the trajectory in this case.

Symplectic Maps Around Elliptic Fixed Points

At an elliptic fixed point, the differential has all its eigenvalues on the unit circle. In this case, a normal form theorem implies that an appropriate, symplectic change of coordinates yield, in a neighborhood of the fixed point, a symplectic twist map of a subset \(U\) of \( T^*T^n\) that is close to being completely integrable (Golé (2001)).

Hamiltonian Diffeomorphisms as Symplectic Twist Maps

Let \(h^t\) be a Hamiltonian flow, associated with the Hamiltonian function \(H\) on the cotangent bundle \(T^*M\ .\) Using the homotopy formula (4) one can show that: \[ (h^t)^*pdq - pdq = dS_t \quad \text{where } S_t(q,p) = \int_\tau pdq - Hdt \] where \(\tau(t) = (h^t(q,p),t)\ ;\) proving that \(h^t\) is exact symplectic. To see when this diffeomorphism satisfies the twist condition, consider the following heuristic argument. For small \(\epsilon\ ,\) we can approximate \(h^\epsilon(q,p) = (q(\epsilon), p(\epsilon))\) by: \[ q(\epsilon) = q + \epsilon H_p + O(\epsilon^2) \] \[ p(\epsilon) = p - \epsilon H_q + O(\epsilon^2) \] The local twist condition \(\det \left(\frac{\partial q(\epsilon)}{\partial p}\right) \neq 0\) is thus equivalent to \(\det H_{pp} \neq 0\) (the Legendre condition). One can make this condition global in different situations; for example, on a compact invariant set or when the non-degeneracy condition on \(H_{pp} \) is suitably uniform. (Golé, 2001). Optimizing \(W(q_0, \ldots , q_N)\) in this case is equivalent to optimizing the Hamiltonian action\(\int pdq - Hdt\) over a set of curves that are piecewise trajectories of \(h^t\) with "corners" at each \(q_k\ .\)

More Relations Between Hamiltonian Diffeomorphisms and Symplectic Twist Maps

Even when no twist condition is available, one can decompose \(h^1\) into a product of symplectic twist maps \(h^1 = (h^{\frac 1N}\circ F)\circ (F^{-1}\circ h^{\frac 1N})\ldots\ ,\) where \(F\) is a symplectic twist map and \(N\) is large enough, yielding a finite variational approach to Hamiltonian trajectories. This method is a generalization of the "broken geodesics" of Morse theory.
(Time independent) Hamiltonian systems also yield exact symplectic maps as return maps on surfaces of section that are hypersurfaces in an energy level. Around a generic elliptic periodic orbits, these maps satisfy a twist condition.
The relation goes the other way: if \(-\partial_{12}S(q,Q)\) is symmetric, definite positive and periodic, it is the generating function of a hamiltonian diffeomorphism.

Periodic Orbits of Symplectic Twist Maps

Given a symplectic map \( F \) on \(T^*M\) that is a composition of symplectic twist maps \( F = F_1\circ\ldots\circ F_K\) (e.g. a Hamiltonian diffeomorphism), one can find periodic orbits as critical points of the corresponding function \[ W(q_0, \ldots , q_{K}) = \sum_{k=0}^{K-1} S_k(q_k, q_{k+1}) \] with suitable periodic boundary conditions, where \(S_k\) is the generating function of \(F_k\ .\) The space of admissible sequences inherits the topology of \(M\ ,\) and hence one finds as many periodic orbit as a function has critical points on \(M\ .\) This requires the appropriate boundary conditions on the dynamics, e.g. that it be that of a geodesic flow outside a bounded set. The boundary condition for a single symplectic twist map can also be given, as proposed by Arnol'd, as a condition of linking of spheres and their images at the boundary of a ball bundle. In the case \(M = T^n\ ,\) one can repeat this procedure to find periodic orbits of all rational rotation vectors \(m/k\) for which \(F^k(q,p)= (q+m,p)\ .\)

Similarly to Floer's proof of the Arnold conjecture, the gradient flow of \(W\) on the space of admissible sequences \((q_0, \ldots, q_K)\) has invariant sets whose (co)homology is at least as complicated as \(H^*(M)\ .\) In the case of \(M =T^n\) these sets are called ghost tori (Golé (2001).

Types of Periodic Orbits. Because of the restrictions on the spectrum of symplectic matrices, periodic orbits of symplectic maps can only be of certain types. In particular they can't have attractive or repulsive periodic orbits. The spectrum of the differential along an orbit is related to the spectrum of the second derivative \(Hess(W)\) of \(W\ :\) in the 2 dimensional case, if an orbit minimizes \(W\) then it is hyperbolic (MacKay & Meiss (1983)). In higher dimensions, the relationship is more complicated: there are open sets of symplectic maps close to integrable that have no hyperbolic fixed points, yet have minimizers (Arnaud (1994)). But given some extra conditions on the generating function, one can relate the hyperbolicity of invariant sets with some uniform non-degeneracy of \(Hess(W)\) (Aubry et. al. (1991)).

Quasiperiodic Orbits

KAM Theory

The completely integrable map \(F_0:(q, p)\mapsto (q + p , p)\) foliates its phase space \(T^*\mathbb T^n\) with invariant tori \(\{ p = p^*\}\ .\) On each torus, \(F_0\) acts as a translation by rotation vector \(p^*\ .\) Each torus is a graph over the base which is Lagrangian (\(\omega \equiv 0\) restricted to each torus). KAM theory states that, for any map \(F\) that is \(C^\infty\) (or even \(C^{2n+1}\)) close to \(F_0\ ,\) the tori with "very" irrational rotation vectors survive as \(F\)-invariant Lagrangian graphs, with dynamics \(C^\infty\)-conjugated to the original one. Very irrational means that \(p^*\) satisfies a Diophantine condition. This condition is shared by a set of large measure made of invariant tori, which tends to full measure as the perturbation goes to 0.

Breakdown of Tori, Chaos and Instability

KAM theory implies that, when the map is close to integrable, with large probability, an orbit belongs to an invariant torus and thus stays bounded for all time. Further theory by Nekhoroshev proves that these tori are sticky: the time that it takes to escape their neighborhood is an exponential function of \(1/\epsilon\ ,\) where \(\epsilon\) is the size of the perturbation. However, the phenomenon of Arnol'd diffusion implies the existence of unbounded orbits in higher dimensional systems. When the symplectic twist map models the dynamics near an elliptic fixed point, this yields instability of the fixed point. R. Douady showed that, near \(F_0\ ,\) there always is a smooth symplectic twist map with Arnold diffusion. Whether these maps are dense in a neighborhood of \(F_0\) is a difficult question which has received partial answers in some Hamiltonian settings; cf. (Mather (1991), de la Llave et. al. (2000)).

For twist maps in dimension 2, break down of invariant circles gives rise to Aubry-Mather (Cantor) invariant sets as well as chaotic orbits that shadow these Cantor sets in any prescribed sequence (Hall (1989)). Chaos can also be explained by the splitting of separatrices, when stable and unstable manifolds of hyperbolic periodic orbits intersect transversally. Variational methods using the generating function can be used to measure the angle of splitting and the rate of transport in phase space.

The existence of orbits of all rotation vectors, guaranteed in dimension 2 by the Aubry-Mather theory, is still hard to establish in higher dimensions, except near the anti-integrable limit (MacKay-Meiss (1992)). There are examples of systems with few rotation directions realized by action minimizing orbits, but little is known about the non-minimizers.

References

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de la Llave, R., Delshams, A. & Seara, T. (2000) A geometric approach to the existence of orbits with unbounded energy in generic periodic perturbations by a potential of generic geodesic flows of \(T^2\ .\)Comm. Math. Phys. 209, no.2

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Internal references

James Meiss (2007) Dynamical systems. Scholarpedia, 2(2):1629.

Ferdinand Verhulst (2007) Hamiltonian normal forms. Scholarpedia, 2(8):2101.

James Meiss (2007) Hamiltonian systems. Scholarpedia, 2(8):1943.

Jeff Moehlis, Kresimir Josic, Eric T. Shea-Brown (2006) Periodic orbit. Scholarpedia, 1(7):1358.

Henk Broer and Heinz Hanßmann (2008) Perturbation theory (dynamical systems). Scholarpedia, 3(9):2399.

Alain Chenciner (2007) Three body problem. Scholarpedia, 2(10):2111.