Dynamics of Hamiltonian Systems

Hanßmann, Heinz

doi:10.1007/978-3-642-27737-5_141-2

Dynamics of Hamiltonian Systems

Heinz Hanßmann²

Living reference work entry
First Online: 01 January 2015

466 Accesses
1 Citations

Definition

Hamiltonian systems are a class of dynamical systems which can be characterized by preservation of a symplectic form. This allows to write down the equations of motion in terms of a single function, the Hamiltonian function. They were conceived in the nineteenth century to study physical systems varying from optics to frictionless mechanics in a unified way. This description turned out to be particularly efficient for symmetry reduction and perturbation analysis.

Introduction

The best-known Hamiltonian system is the harmonic oscillator. The second-order differential equation

$$ \ddot{x}+{\varpi}^2x=0 $$

models the motion of a point mass attached to a massless spring (Hooke’s law) and has the general solution

$$ x(t)={x}_0 \cos \varpi t+\frac{y_0}{\varpi } \sin \varpi t $$

with initial conditions $ \left(x(0),\dot{x}(0)\right)=\left({x}_0,{y}_0\right) $. Choosing co-ordinates

$$ \begin{array}{lll}q=\sqrt{\varpi x}\hfill & \mathrm{and}\hfill & p=\frac{y}{\sqrt{\varpi...

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Abbreviations

Action angle variables:

In an integrable Hamiltonian system with n degrees of freedom, the level sets of regular values of the n integrals are Lagrangean submanifolds. In case these level sets are compact, they are (unions of) n-tori. In the neighborhood of such an invariant torus, one can find a Darboux chart with co-ordinates (x, y) such that the integrals (and in particular the Hamiltonian) depend only on the actions y. The values of (x, y) form a product set $ {\mathbb{T}}^n\times \mathbb{Y} $, with $ \mathbb{Y}\subseteq {\mathrm{\mathbb{R}}}^n $ open. Every invariant n-torus y = const in the domain of this chart is parametrized by the angles x.

Canonical 1-form:

The symplectic form (or canonical 2-form) is closed, satisfying dω = 0. If it is furthermore exact, i.e., of the form $ \omega =\mathrm{d}\theta $, then $ \vartheta =-\theta $ is called a canonical 1-form (unique up to addition of df, $ f\in {C}^{\infty}\left(\mathcal{P}\right) $).

Canonical transformation:

An invertible transformation that preserves the Poisson bracket, turning canonical co-ordinates into canonical co-ordinates. In the symplectic case, it is also called symplectomorphism.

Cantor set, Cantor dust, Cantor family, and Cantor stratification:

Cantor dust is a separable locally compact space that is perfect, i.e., every point is in the closure of its complement, and totally disconnected. This determines Cantor dust up to homeomorphy. The term Cantor set (originally reserved for the specific form of Cantor dust obtained by repeatedly deleting “the middle third”) designates topological spaces that locally have the structure $ {\mathrm{\mathbb{R}}}^n\times Cantor\kern0.5em dust $ for some $ n\in \mathrm{\mathbb{N}} $. Cantor families are parametrized by such Cantor sets. On the real line ℝ, one can define Cantor dust of positive measure by excluding around each rational number p/q an interval of size 2γ/q ^τ, $ \gamma >0 $, and $ \tau >2 $. Similar Diophantine conditions define Cantor sets in ℝⁿ. Since these Cantor sets have positive measure, their (Hausdorff)-dimension is n. Where the unperturbed system is stratified according to the codimension of occurring (bifurcating) tori, this leads to a Cantor stratification.

Casimir function:

A function $ f:\mathcal{P}\to \mathrm{\mathbb{R}} $ is a Casimir element of the Poisson algebra $ {C}^{\infty}\left(\mathcal{P}\right) $ if {f, g} = 0 for all $ g\in {C}^{\infty}\left(\mathcal{P}\right) $. This induces a Poisson structure on the level sets $ {f}^{-1}(a) $ and $ a\in \mathrm{\mathbb{R}} $. In case every point has a small neighborhood on which the only Casimir elements are the constant functions, the Poisson structure on $ \mathcal{P} $ is nondegenerate, i.e., $ \mathcal{P} $ is a symplectic manifold.

Center:

An equilibrium of a vector field is called a center if all eigenvalues of the linearization are nonzero and on the imaginary axis.

Center-saddle bifurcation:

Under variation of a parameter λ, a center and a saddle meet and vanish.

Chern class:

Let $ \rho :\mathrm{\mathcal{R}}\to C $ be a torus bundle with fiber $ {\mathbb{T}}^n $ and denote by $ \mathcal{G} $ the locally constant sheaf of first homotopy groups $ {\pi}_1\left({\mathbb{T}}^n\right) $ of the fibers. The Chern class of the torus bundle is an element of $ {H}^2\left(C,\mathcal{G}\right) $ that measures the obstruction to the existence of a global section of ρ – such a section exists if and only if the Chern class vanishes. An example with a nonvanishing Chern class is the Hopf fibration $ {S}^3\to {S}^2 $.

Conditionally periodic:

A motion $ t\mapsto \alpha (t)\in \mathcal{P} $ is conditionally periodic if there are frequencies $ {\varpi}_1,\dots, {\varpi}_k\in \mathrm{\mathbb{R}} $ and a smooth embedding $ F:{\mathbb{T}}^k\to \mathcal{P} $ such that $ \alpha (t)=F\left({\mathrm{e}}^{2\pi \mathrm{i}{\varpi}_1t},\dots, {e}^{2\pi \mathrm{i}{\varpi}_kt}\right) $. One can think of the motion as a superposition of the periodic motions $ t\mapsto F\left(1,\dots, 1,{\mathrm{e}}^{2\pi \mathrm{i}{\varpi}_jt},1,\dots, 1\right) $. If the frequencies are rationally independent, the motion $ t\mapsto \alpha (t)\in \mathcal{P} $ lies dense in imF, and this embedded torus is invariant. In case there are resonances among the frequencies, the motion is restricted to a subtorus.

A flow on a torus is parallel or conditionally periodic if there exist co-ordinates in which the vector field becomes constant. In the absence of resonances, the flow is called quasi-periodic.

Conjugate co-ordinates and canonical co-ordinates:

Two co-ordinates Q and P of a Darboux chart are (canonically) conjugate if $ \omega \left({X}_Q,{X}_P\right)=\pm 1 $, i.e., X _Q and X _P span for every point x in the domain U of the chart a hyperbolic plane in T _x U.

Darboux chart, Darboux co-ordinates, and Darboux basis:

In a Darboux basis {e ₁,…, e _n, f ₁,…, f _n} of a symplectic vector space (V, ω), the symplectic product takes the simple form $ \omega \left({e}_i,{e}_j\right)=0=\omega \left({f}_i,{f}_j\right) $ and $ \omega \left({e}_i,{f}_j\right)={\delta}_{ij} $. In Darboux co-ordinates (q ₁, …, q _n, p ₁, …, p _n) of a symplectic manifold $ \left(\mathcal{P},\omega \right) $, the symplectic form becomes $ \omega ={\displaystyle \sum \mathrm{d}{q}_i\wedge \mathrm{d}{p}_i} $.

Degree of freedom:

In simple mechanical systems, the phase space is the cotangent bundle of the configuration space, and the dimension of the latter encodes “in how many directions the system can move.” For symplectic manifolds, this notion is immediately generalized to one half of the dimension of the phase space. Poisson spaces are foliated by their symplectic leaves, and the number of degrees of freedom is defined to be one half of the rank of the Poisson structure.

Diophantine condition and Diophantine frequency vector:

A frequency vector $ \varpi \in {\mathrm{\mathbb{R}}}^n $ is called Diophantine if there are constants γ > 0 and τ > n − 1 with $ \begin{array}{ll}\left|\left\langle k\Big|\varpi \right\rangle \right|\ge \frac{\gamma }{\left|k\right|{}^{\tau }}\hfill & \mathrm{f}\mathrm{o}\mathrm{r}\ \mathrm{all}\kern0.5em k\in {\mathbb{Z}}^{n\backslash \left\{0\right\}}\hfill \end{array}. $

The Diophantine frequency vectors satisfying this condition for fixed γ and τ form a Cantor set of half lines. As the “Diophantine parameter” γ tends to zero (while τ remains fixed), these half lines extend to the origin. The complement in any compact set of frequency vectors satisfying a Diophantine condition with fixed τ has a measure of order O(γ) as γ → 0.

Elliptic:

A periodic orbit/invariant torus is elliptic if all Floquet multipliers/exponents are on the unit circle/imaginary axis. An elliptic equilibrium is called a center.

Energy shell:

The set of points that can be attained given a certain energy. If the phase space is a symplectic manifold, this set is given by the pre-image of that energy value under the Hamiltonian. For more general Poisson spaces, this pre-image has to be intersected with a symplectic leaf.

Focus-focus type:

The (normal) linear behavior generated by a quartet of eigenvalues/Floquet multipliers/exponents.

Frequency-halving bifurcation:

In the supercritical case, an invariant torus loses its stability, giving rise to a stable torus with one of its frequencies halved. In the dual or subcritical case, stability is lost through collision with an unstable invariant torus with one of its frequencies halved. The periodic counterpart is called a period-doubling bifurcation.

Generating function:

A function S = S(q, y) that defines a canonical transformation $ \left(x,y\right)\mapsto \left(q,p\right) $ by means of $ x=\left(\partial S\right)/\left(\partial y\right) $ and $ p=\left(\partial S\right)/\left(\partial q\right) $. The generating function always depends on both the old and the new variables, working with dependence on, e.g., x instead of y introduces a minus sign.

Group action:

A mapping $ \Gamma :G\times \mathcal{P}\to \mathcal{P} $ is a group action of the Lie group G on the phase space $ \mathcal{P} $ if $ {\Gamma}_e=\mathrm{id} $ and $ {\Gamma}_{g\cdot h}={\Gamma}_g\circ {\Gamma}_h $ for all Lie group elements g and h. For $ G=\mathrm{\mathbb{R}} $, one can recover the generating vector field by means of $ X=\frac{\mathrm{d}}{\mathrm{d}t}{\Gamma}_t $.

Hamiltonian Hopf bifurcation:

In a two-degree-of-freedom system, a center loses its stability: the two pairs of purely imaginary eigenvalues meet in a Krein collision. In a three-degree-of-freedom system, the same can happen to two pairs $ {\mathrm{e}}^{\pm \lambda } $ and $ {\mathrm{e}}^{\pm \mu } $ of Floquet multipliers in a one-parameter family of periodic orbits – parametrized by the energy. For a quasi-periodic Hamiltonian Hopf bifurcation, one needs at least four degrees of freedom.

Hamiltonian system:

Newton’s second law states $ F=m\ddot{q} $. Suppose that F is a conservative force, with potential V. Write the equations of motion as a system of first-order differential equations

$$ \dot{q}=\frac{1}{m}p $$

$$ \dot{p}=-\frac{\partial V}{\partial q} $$

that has the total energy $ H\left(q,p\right)=\left\langle p\Big|p\right\rangle /(2m)+V(q) $ as a first integral. This can be generalized. Given a Hamiltonian function H(q, p), one has the Hamiltonian vector field

$$ \begin{array}{c}\hfill \dot{q}=\frac{\partial H}{\partial p}\hfill \\ {}\hfill \dot{p}=-\frac{\partial H}{\partial q}\hfill \end{array} $$

with first integral H. Moreover, one can replace ℝ²ⁿ by a symplectic manifold or by a Poisson space.

Hilbert basis:

The (smooth) invariants of an action of a compact group are all given as functions of finitely many “basic” invariants.

Hypoelliptic:

An equilibrium is called hypoelliptic if its linearization has both elliptic and hyperbolic eigenvalues, all nonzero. Similar for periodic orbits and invariant tori.

Integrable system:

A Hamiltonian system with n degrees of freedom is (Liouville)-integrable if it has n functionally independent commuting integrals of motion. Locally, this implies the existence of a (local) torus action.

Integral (of motion):

A conserved quantity.

Iso-energetic Poincaré mapping:

For a Hamiltonian system, a Poincaré mapping leaves the energy shells invariant. Restricting to the intersection $ \varSigma \cap \left\{H=h\right\} $ of the Poincaré section with an energy shell, one obtains an iso-energetic Poincaré mapping.

Isotropic:

For a subspace U < V of a symplectic vector space (V, ω) are equivalent: (i) U is contained in its ω-orthogonal complement, (ii) every basis of U can be extended to a Darboux basis of V. If U satisfies one and thus both of these conditions, it is called an isotropic subspace. A submanifold $ \mathcal{B}\subseteq \mathcal{P} $ of a symplectic manifold $ \left(\mathcal{P},\omega \right) $ is called isotropic if all tangent spaces $ {T}_x\mathcal{B}<{T}_x\mathcal{P} $ are isotropic subspaces.

Krein collision:

Two pairs of purely imaginary Floquet exponents meet in a double pair on the imaginary axis and split off to form a complex quartet $ \pm \mathrm{\Re}\pm \mathrm{i}\Im . $ This is a consequence of a transversality condition on the linear terms at 1:–1 resonance; an additional nondegeneracy condition on the nonlinear part ensures that a ((quasi)-periodic) Hamiltonian Hopf bifurcation takes place.

Lagrangean submanifold:

For a subspace U < V of a symplectic vector space (V, ω) are equivalent: (i) U is a maximal isotropic subspace, (ii) V has a Darboux basis $ \left\{{e}_1,\dots, {e}_n,{f}_1,\dots, {f}_n\right\} $ with span $ \left\{{e}_1,\dots, {e}_n\right\}=U $. If U satisfies one and thus both of these conditions, it is called a Lagrangean subspace. A submanifold $ \mathcal{B}\subseteq \mathcal{P} $ of a symplectic manifold $ \left(\mathcal{P},\omega \right) $ is called Lagrangean if all tangent spaces $ {T}_x\mathcal{B}<{T}_x\mathcal{P} $ are Lagrangean subspaces.

Lie–Poisson structure:

Let $ \mathfrak{g} $ be a Lie algebra with structure constants Γ ^k_ij . Then $ \left\{{\mu}_i,{\mu}_j\right\}=\pm {\displaystyle \sum {\Gamma}_{ij}^k{\mu}_k} $ defines two Poisson structures on the dual space $ \mathfrak{g}* $.

Local bifurcation:

Bifurcations of equilibria can be studied within a small neighborhood in the product of phase space and parameter space. The same is true for fixed points of a discrete dynamical system, but when suspended to a flow, the corresponding bifurcating periodic orbits obtain a semi-local character.

Momentum mapping:

Let $ \Gamma :G\times \mathcal{P}\to \mathcal{P} $ be a symplectic action of the Lie group G on the symplectic manifold $ \mathcal{P} $. A mapping $ J:\mathcal{P}\to \mathfrak{g}* $ into the dual space of the Lie algebra of G is a momentum mapping for the action if $ {X}_{J^{\xi }}={\xi}_{\mathcal{P}} $ for all $ \xi \in \mathfrak{g} $. Here, $ {J}^{\xi }:\mathcal{P}\to \mathrm{\mathbb{R}} $ is defined by $ {J}^{\xi }(z)=J(z)\cdot \xi $ and $ {\xi}_{\mathcal{P}} $ is the infinitesimal generator of the action corresponding to ξ. The momentum mapping J is called Ad*-equivariant provided that $ J\circ {\Gamma}_g={\mathrm{Ad}}_{g^{-1}}^{*}\circ J $.

Monodromy:

The group homomorphism $ \mathrm{\mathcal{M}}:{\pi}_1(C)\to S{L}_n\left(\mathrm{\mathbb{Z}}\right) $ that measures how much the n-torus bundle $ \mathrm{\mathcal{R}}\to C $ deviates from a principal torus bundle.

Nondegenerate integrable Hamiltonian system, function:

In action angle variables (x, y), the Hamiltonian H only depends on the action variables y, and the equations of motion become $ \dot{x}=\varpi (y) $ and $ \dot{y}=0 $, with frequencies $ \varpi (y)=\left(\partial H\right)/\left(\partial y\right) $. The system is nondegenerate at the invariant torus {y = y ₀} if D ² H(y ₀) is invertible. In this case, ϖ defines near {y = y ₀} an isomorphism between the actions and the angular velocities. Other conditions ensuring that most tori have Diophantine frequency vectors are iso-energetic nondegeneracy or Rüssmann-like conditions on higher derivatives.

Normal frequency:

Given an elliptic invariant torus of a Hamiltonian system, one can define the normal linearization on the symplectic normal bundle. The eigenvalues of the normal linearization being $ \pm {\mathrm{i}\Omega}_1,\dots, \pm {\mathrm{i}\Omega}_m $, one calls the Ω_j the normal frequencies. Under the exponential mapping, the eigenvalues of the normal linearization of a periodic orbit are mapped to Floquet multipliers.

Normal linearization:

The linearization within the normal bundle of an invariant submanifold. In the Hamiltonian context, these are often isotropic, and one further restricts to the symplectic normal linear behavior, e.g., to identify elliptic and hyperbolic invariant tori.

Normal modes:

A family of periodic orbits parametrized by the energy value h, as h → h ₀ shrinking down to an equilibrium that has a pair of eigenvalues $ \pm 2\pi \mathrm{i}\underset{h\to {h}_0}{ \lim }{T}_h^{-1} $, where T _h denotes the period(s).

Parabolic:

An equilibrium of a one-degree-of-freedom system is called parabolic if its linearization is nilpotent but nonzero. An invariant torus is parabolic if its symplectic normal linearization has a parabolic equilibrium. In particular, the four Floquet multipliers of a parabolic periodic orbit in two degrees of freedom are all equal to 1.

Phase space:

By Newton’s second law, the equations of motion are second-order differential equations. The trajectory is completely determined by the initial positions and the initial velocities, or, equivalently, the initial momenta. The phase space is the set of all possible combinations of initial positions and initial momenta.

Pinched torus:

The compact (un)stable manifold of a saddle in two degrees of freedom with a quartet $ \pm \Re \pm \mathrm{i}\Im $ of hyperbolic eigenvalues resembles a torus $ {\mathbb{T}}^2={S}^1\times {S}^1 $ with one of the fibers $ \left\{x\right\}\times {S}^1 $ reduced to a point.

Poisson space, Poisson bracket, Poisson structure:

A Poisson algebra $ \mathcal{A} $ is a real Lie algebra that is also a commutative ring with unit. These two structures are related by Leibniz’ rule $ \left\{f\cdot g,h\right\}=f\cdot \left\{g,h\right\}+g\cdot \left\{f,h\right\} $. A Poisson manifold $ \mathcal{P} $ has a Poisson bracket on $ {C}^{\infty}\left(\mathcal{P}\right) $ that makes $ {C}^{\infty}\left(\mathcal{P}\right) $ a Poisson algebra. If there are locally no Casimir elements other than constant functions, this leads to a symplectic structure on $ \mathcal{P} $.

Poisson spaces naturally arise in singular reduction; this motivates to allow varieties $ \mathcal{P} $ where the Poisson bracket is defined on a suitable subalgebra $ \mathcal{A} $ of $ C\left(\mathcal{P}\right) $.

Given a Hamiltonian function $ H\in \mathcal{A} $, one obtains for $ f\in \mathcal{A} $ the equations of motion $ \frac{\mathrm{d}}{\mathrm{d}t}f=\left\{f,H\right\} $. For canonically conjugate co-ordinates (q, p) on $ \mathcal{P} $, i.e., with $ \left\{{q}_i,{q}_j\right\}=0=\left\{{p}_i,{p}_j\right\} $ and $ \left\{{q}_i,{p}_j\right\}={\delta}_{ij} $, this amounts to

$$ \begin{array}{c}\hfill \dot{q}=\frac{\partial H}{\partial p}\hfill \\ {}\hfill \dot{p}=-\frac{\partial H}{\partial q}.\hfill \end{array} $$

Poisson symmetry:

A symmetry that preserves the Poisson structure.

Principal fiber bundle:

A fiber bundle $ F:\mathcal{P}\to C $ admitting a free Lie group action $ G\times \mathcal{P}\to \mathcal{P} $ that acts transitively on the fibers $ {F}^{-1}(c)\cong G $.

Proper degeneracy:

For the application of the KAM theorem to a perturbation of an integrable system, it is necessary that the integrable system is nondegenerate, so that the frequencies have maximal rank as function of the actions. If there are global conditions relating the frequencies, so that the conditionally periodic motion can be described by a smaller number of these, the system is properly degenerate. Superintegrable systems are a particular example.

Quasi-periodic:

A conditionally periodic motion that is not periodic. The closure of the trajectory is an invariant k-torus with k ≥ 2.

A parallel or conditionally periodic flow on a k-torus is called quasi-periodic if the frequencies ϖ ₁, …, ϖ _k are rationally independent.

Ramified torus bundle:

Let a differentiable mapping $ f:\mathcal{P}\to {\mathrm{\mathbb{R}}}^m $ be given. According to Sard’s Lemma, almost all values $ a\in {\mathrm{\mathbb{R}}}^m $ are regular. The connected components of the sets $ {f}^{-1}(a) $ and $ a\in \mathrm{i}\mathrm{m}\;f $ regular, define a foliation of an open subset ℛ of the (n + m)-dimensional manifold $ \mathcal{P} $.

In the present settings, the components of f are the first integrals of an integrable Hamiltonian system with compact level sets. Then the regular fibers are n-tori, and their union is a torus bundle. The topology of this bundle depends on the topology of the base space C, the monodromy, and the Chern class of the bundle. In many examples that one encounters, the connected components of C are contractible, whence monodromy and Chern classes are trivial.

For the geometry of the bundle, one also wants to know how the singular fibers are distributed: where n-tori shrink to normally elliptic (n − 1)-tori and where they are separated by stable and unstable manifolds of normally hyperbolic (n − 1)-tori. The singular fibers are not necessarily manifolds but may be stratified into X _H-invariant strata which are possibly noncompact.

One can continue and look for the singularities of these “regular singular leaves,” the tori of dimension ≤n − 2 and the normally parabolic (n − 1)-tori, in which normally elliptic and normally hyperbolic (n − 1)-tori meet in a quasi-periodic center-saddle bifurcation or a frequency-halving bifurcation. The next “layer” is given by quasi-periodic Hamiltonian Hopf bifurcations and bifurcations of higher codimension.

Reduced phase space:

Let $ \Gamma :G\times \mathcal{P}\to \mathcal{P} $ be a symplectic action of the Lie group G on the symplectic manifold $ \mathcal{P} $ with Ad*-equivariant momentum mapping $ J:\mathcal{P}\to \mathfrak{g}* $. For a regular value $ \mu \in \mathfrak{g}* $, let the action of the isotropy group $ {G}_{\mu }=\left\{g\in G\Big|{\mathrm{Ad}}_{g^{-1}}^{*}\left(\mu \right)=\mu \right\} $ on $ {J}^{-1}\left(\mu \right) $ be free and proper. Then the quotient $ {J}^{-1}\left(\mu \right)/{G}_{\mu } $ is again a symplectic manifold, the reduced phase space. A Γ-invariant Hamiltonian function H on $ \mathcal{P} $ leads to a reduced Hamiltonian function H _μ on the reduced phase space.

Relative equilibrium:

Let H be a Hamiltonian function invariant under the symplectic action $ G\times \mathcal{P}\to \mathcal{P} $, and let $ \mu \in \mathfrak{g}* $ be a regular value of the Ad*-equivariant momentum mapping $ J:\mathcal{P}\to \mathfrak{g}* $. Also assume that the isotropy group G _μ under the Ad* action on $ \mathfrak{g}* $ acts freely and properly on $ {J}^{-1}\left(\mu \right) $. Then X _H induces a Hamiltonian flow on the reduced phase space $ {\mathcal{P}}_{\mu }={J}^{-1}\left(\mu \right)/{G}_{\mu } $. The phase curves of the given Hamiltonian system on $ \mathcal{P} $ with momentum constant $ J=\mu $ that are taken by the projection $ {J}^{-1}\left(\mu \right)\to {\mathcal{P}}_{\mu } $ into equilibrium positions of the reduced Hamiltonian system are called relative equilibria or stationary motions (of the original system).

Remove the degeneracy:

A perturbation of a superintegrable system removes the degeneracy if it is sufficiently mild to define an intermediate system that is still integrable and sufficiently wild to make that intermediate system nondegenerate.

Resonance:

If the frequencies of an invariant torus with conditionally periodic flow are rationally dependent, this torus divides into invariant subtori. Such resonances $ \left\langle h\Big|\varpi \right\rangle =0 $ and $ h\in {\mathrm{\mathbb{Z}}}^k $ define hyperplanes in ϖ space and, by means of the frequency mapping, also in phase space. The smallest number $ \left|h\right|=\left|{h}_1\right|+\cdots +\left|{h}_k\right| $ is the order of the resonance. Diophantine conditions describe a measure-theoretically large complement of a neighborhood of the (dense!) set of all resonances.

Saddle:

An equilibrium of a vector field is called a saddle if the linearization has no eigenvalues on the imaginary axis. On a small neighborhood of a saddle, the flow is topologically conjugate to its linearization.

Semi-local bifurcation:

Bifurcations of n-tori can be studied in a tubular neighborhood. For n = 1, a Poincaré section turns the periodic orbit into a fixed point of the Poincaré mapping, and the bifurcation obtains a local character.

Simple mechanical system:

A quintuple $ \left({T}^{*}M,\left\langle ..\Big|..\right\rangle, V,T,\omega \right) $ consisting of the cotangent bundle of a Riemannian manifold $ \left(M,\left\langle ..\Big|..\right\rangle \right) $, a potential function $ V:M\to \mathrm{\mathbb{R}} $, the kinetic energy $ T\left(\alpha \right)=\left\langle \alpha \Big|\alpha \right\rangle $, and the symplectic form $ \omega =-\mathrm{d}\vartheta $ derived from the canonical 1-form on the cotangent bundle T*M.

Singular reduction:

If $ \Gamma :G\times \mathcal{P}\to \mathcal{P} $ is a Poisson symmetry, then the group action on $ \mathcal{P} $ makes $ \mathcal{B}=\mathcal{P}/G $ a Poisson space as well. Fixing the values of the resulting Casimir functions yields the reduced phase space, which turns out to have singular points where the action Γ is not free.

Solenoid:

Given a sequence $ {f}_j:{S}^1\to {S}^1 $ of coverings $ {f}_j\left(\zeta \right)={\zeta}^{\alpha_j} $ of the circle S ¹, the solenoid $ {\varSigma}_a\subseteq {\left({S}^1\right)}^{{\mathrm{\mathbb{N}}}_0} $, $ a={\left({\alpha}_j\right)}_{j\in {\mathrm{\mathbb{N}}}_0} $ consists of all $ z={\left({\zeta}_j\right)}_{j\in {\mathrm{\mathbb{N}}}_0} $ with $ {\zeta}_j={f}_j\left({\zeta}_{j+1}\right) $ for all $ j\in {\mathrm{\mathbb{N}}}_0 $.

Stratification:

The decomposition of a topological space into smaller pieces satisfying certain boundary conditions.

Superintegrable system:

A Hamiltonian system with n degrees of freedom is superintegrable if it has n + 1 functionally independent integrals of motion such that each of the first n − 1 of them commutes with all n + 1. Such a properly degenerate system admits generalized action angle co-ordinates $ \left({x}_1,\dots, {x}_{n-1},{y}_1,\dots, {y}_{n-1},q,p\right) $. In case the nondegeneracy condition det $ {D}^2H(y)\ne 0 $ is satisfied almost everywhere, the system is “minimally superintegrable.” In the other extreme of a “maximally superintegrable” system, all motions are periodic.

Symplectic manifold:

A 2n-dimensional manifold $ \mathcal{P} $ with a nondegenerate closed two-form ω, i.e., dω = 0 and ω(u, v) = 0 for all $ v\in T\mathcal{P}\Rightarrow u=0 $. A diffeomorphism ψ of symplectic manifolds that respects the two-form(s) is called a symplectomorphism. Given a Hamiltonian function $ H\in {C}^{\infty}\left(\mathcal{P}\right) $, one obtains through $ \omega \left({X}_H,\dots \right)=\mathrm{d}H $ the Hamiltonian vector field $ {X}_H $. For every $ x\in \mathcal{P} $, there are co-ordinates (q, p) around x with $ \omega =\mathrm{d}q\wedge \mathrm{d}p $. In these Darboux co-ordinates $ {X}_H $ reads

$$ \begin{array}{c}\hfill \dot{q}=\frac{\partial H}{\partial p}\hfill \\ {}\hfill \dot{p}=-\frac{\partial H}{\partial q}.\hfill \end{array} $$

Symplectic form:

A nondegenerate closed two-form.

Syzygy:

A constraining equation that is identically fulfilled by the elements of a Hilbert basis.

(Un)stable manifold:

In Hamiltonian systems with one degree of freedom, the stable manifold and the unstable manifold of an equilibrium often coincide and thus consist of homoclinic orbits. In such a case, it is called an (un)stable manifold. This carries over to the stable and the unstable manifold of a periodic orbit or an invariant torus in higher degrees of freedom if the system is integrable.

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Acknowledgments

The author thanks Henk Broer, Marius Crainic, Hans Duistermaat, Luuk Hoevenaars, Hessel Posthuma, and Ferdinand Verhulst for fruitful discussions and helpful remarks.

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Mathematisch Instituut, Universiteit Utrecht, 3508 TA, Utrecht, The Netherlands
Heinz Hanßmann

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Heinz Hanßmann
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Robert A. Meyers

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Hanßmann, H. (2015). Dynamics of Hamiltonian Systems. In: Meyers, R. (eds) Encyclopedia of Complexity and Systems Science. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-27737-5_141-2

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DOI: https://doi.org/10.1007/978-3-642-27737-5_141-2
Received: 27 January 2015
Accepted: 27 January 2015
Published: 24 February 2015
Publisher Name: Springer, Berlin, Heidelberg
Online ISBN: 978-3-642-27737-5
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