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Kolmogorov-Arnold-Moser theory
Luigi Chierchia and John N. Mather (2010), Scholarpedia, 5(9):2123. | doi:10.4249/scholarpedia.2123 | revision #91405 [link to/cite this article] |
Kolmogorov-Arnold-Moser (KAM) theory deals with persistence, under perturbation, of quasi-periodic motions in Hamiltonian dynamical systems.
An important example is given by the dynamics of nearly-integrable Hamiltonian systems. In general, the phase space of a completely integrable Hamiltonian system of \(n\) degrees of freedom is foliated by invariant \(n\)-dimensional tori (possibly of different topology). KAM theory shows that, under suitable regularity and non-degeneracy assumptions, most (in measure theoretic sense) of such tori persist (slightly deformed) under small Hamiltonian perturbations. The union of persistent \(n\)-dimensional tori (Kolmogorov set) tend to fill the whole phase space as the strength of the perturbation is decreased.
The major technical problem arising in this context is due to the appearance of resonances and of small divisors in the associated formal perturbation series.
Contents |
Classical KAM theory
The main objects studied in KAM theory are \(d\)-dimensional embedded tori \(\mathcal{T}^d\) invariant for a Hamiltonian flow \(\phi^t_H: \mathcal{M}^{2n}\to\mathcal{M}^{2n}\ ,\) where \(t\in\mathbb{R}\) denotes the time variable and \(H=H(p,q)\) is a (smooth enough or analytic) Hamiltonian function depending on \(2n\) symplectic (or canonical) variables \(p=(p_1,...,p_n)\) and \(q=(q_1,...,q_n)\) defined on the phase space \(\mathcal{M}^{2n}\ .\) This means that if \((p_0,q_0)\in\mathcal{T}^d\ ,\) then \(\phi^t_H(p_0,q_0)\in\mathcal{T}^d\) for any \(t\in\mathbb{R}\ ,\) \(\phi^t_H(p_0,q_0)=(p(t),q(t))\) denoting the solution of the (standard) Hamilton equations \[\tag{1} \left\{\begin{array}{l}\dot p = - \partial_q H(p,q)\\ \dot q = \partial_p H(p,q)\end{array}\right.\quad{\rm with\ initial\ data }\quad \left\{\begin{array}{l} p(0)=p_0\\ q(0)=q_0\end{array}\right. . \]
Here, the dot represents time derivative, while \(\partial_z\) denotes the gradient with respect to the \(z\) variables.
A \(d\)-dimensional (embedded and smooth or analytic) invariant torus for \(\phi_H^t\ ,\) with \(2\le d\le n\ ,\) is called a KAM torus if:
- the flow \(\phi^t_H\) on \(\mathcal{T}^d\) is conjugated to a linear translation \(\theta \to \theta + \omega t\ ,\) where \(\theta=(\theta_1,...,\theta_d)\) belongs to the standard \(d\)-dimensional torus \(\mathbb{T}^d=\mathbb{R}^d/(2\pi \mathbb{Z})^d\ ;\) the vector \(\omega=(\omega_1,...,\omega_d)\in\mathbb{R}^d\) is called the frequency vector;
- the frequency vector \(\omega\) is rationally independent and "badly" approximated by rationals, typically in a Diophantine sense:
\[\tag{2} \exists\ \gamma, \tau>0\ {\rm such\ that} \quad |\omega\cdot k|:= |\sum_{j=1}^d \omega_j k_j|\ge \frac{\gamma}{\|k\|^\tau}\ ,\ \forall\ k\in\mathbb{Z}^d\backslash\{0\}\ . \]
From measure theory, it follows that the set of Diophantine vectors in \(\mathbb{R}^d\) is of full Lebesgue measure.
Note that the case \(d=1\) corresponds to periodic trajectories of period \(2\pi/\omega\) (this case is normally excluded in classical KAM theory since does not involve small divisors). On the other hand, the case \(d=n\) corresponding to maximal KAM tori is particularly relevant.
Kolmogorov normal forms and Kolmogorov's Theorem
Let \(H\) be a real-analytic Hamiltonian on \(\mathcal{M}^{2n}=U\times \mathbb{T}^n\) (with \(U\) an open region in \(\mathbb{R}^n\)) and assume that \(\mathcal{T}^n\) is a maximal KAM torus for \(H\) and that it is a (Lagrangian) graph over the angle variables. Then there exists a symplectic transformation \(\phi: (y,x)\to(p,q)\) (i.e., a diffeomorphism preserving the canonical 2-form \(\displaystyle \sum_{i=1}^n dp_i\wedge dq_i\)) transforming \(H\) in Kolmogorov normal form: \[\tag{3} H\circ\phi(y,x)=K(y,x):=E+\omega\cdot y + Q(y,x) \]
for some number \(E\) (the energy level of the KAM torus), some Diophantine frequency vector \(\omega\) and \(Q\) a function vanishing together with its first \(y\)-derivatives at \(y=0\ .\) In the "new" variables \((y,x)\ ,\) the \(n\)-torus \(\{0\}\times\mathbb{T}^n\) is obviously a KAM torus for the transformed Hamiltonian \(H\circ\phi\ .\)
One says that the Kolmogorov normal form \(K\) in (3) is non-degenerate if the Hessian matrix (with respect to \(y\)) of the average of \(Q\) over \(\mathbb{T}^n\) is regular.
Kolmogorov's Theorem (Kolmogorov, 1954) Let \(K\) be a real-analytic non-degenerate Kolmogorov's normal form and let \(P\) be a real-analytic function in a neighborhood of \(\{y=0\}\times\mathbb{T}^n\ .\) Then, there exists \(\epsilon_0>0\) and for any \(|\epsilon|<\epsilon_0\) a real-analytic symplectic transformation \(\phi_\epsilon\ ,\) close to the identity, such that, if \(H_\epsilon\) denotes the perturbed Hamiltonian \(K+\epsilon P\ ,\) then \(H_\epsilon\circ\phi_\epsilon\) is a non-degenerate Kolmogorov's normal form with the same frequency vector of \(K\ .\)
Thus, in particular, \(\mathcal{T}_\epsilon^n:=\phi_\epsilon(\{0\}\times \mathbb{T}^n)\) is a (real-analytic, non-degerate) KAM torus for \(H_\epsilon\) and such a torus is \(\epsilon\)-close to \(\{0\}\times\mathbb{T}^n\ .\)
Nearly-integrable Hamiltonian systems
A nearly-integrable Hamiltonian system is a Hamiltonian system governed by a Hamiltonian function of the form \(H_\epsilon(y,x)=K(y)+\epsilon P(y,x)\) with \(y=(y_1,...,y_n)\) (action variables) varying in a domain \(B\subset \mathbb{R}^n\) and \(x=(x_1,...,x_n)\) (angle variables) varying in the standard \(n\)-dimensional torus \(\mathbb{T}^n\ .\) For \(\epsilon=0\ ,\) equations (1) give \(\dot y=0\) and \(\dot x=\partial_y K(y)\ ,\) hence \(y=y_0=\) constant and \(x = x_0 + \omega_0\, t\) (mod \(2\pi\)), with \(\omega_0:= \partial_y K\big(y_0\big)\ .\) Thus the torus \(\{y_0\}\times \mathbb{T}^n\) is invariant for the flow \(\phi^t_{K}\ ,\) and if \(\omega_0\) is Diophantine and \(\partial_y^2 K(y_0)\) is invertible, then such a torus is a non-degenerate KAM torus for \(H_0=K\ .\) Since \(K(y)\) can be expanded by Taylor's formula as \[ K=K(y_0)+\omega_0\cdot (y-y_0)+ \frac{1}{2} \partial_y^2 K(y_0) (y-y_0) \cdot(y-y_0)+ O(|y-y_0|^3|), \] it follows from Kolmogorov's Theorem that for \(\epsilon\) small enough such tori persist, giving rise to non-degenerate KAM tori for \(H_\epsilon\).
Moser's differentiable version
J.K. Moser (followed by H. Rüssmann, J. Pöschel and others) showed that the real-analyticity assumption is not necessary. Indeed, Kolmogorov's Theorem holds under the milder assumption that \(H\) is a \(C^\ell\) differentiable function with \(\ell>2n\) (meaning that \(H\) is of class \(C^{2n}\) and that the derivatives of order \(2n\) are Hölder continuous).
Originally (Moser, 1962), Moser's work focused on \(C^{333}\) (exact symplectic) perturbations of integrable twist mappings of the annulus (the most famous example being the so-called standard map). In this case, maximal KAM tori correspond to homotopically non-trivial curves intersecting each radius in only one point. The number of derivatives were reduced to 5 by H. Rüssmann (Rüssmann, 1970), and M. Herman (Herman, 1983) showed that the theorem is valid for \(C^k\) perturbation with \(k\ge 3\ ,\) but false for \(k<3\ .\)
Small divisors and classical KAM techniques
KAM techniques (i.e., the analytical tools used to prove statements in KAM theory) constitute the hard core of KAM theory and play a major role in applications, extensions and, in general, in the full comprehension of the results. The main technical problem is related to the appearance of small divisors in the Fourier series of perturbative expansions (averaging methods, series expansions of quasi-periodic motions, etc.).
Small divisors are expressions of the form \(\omega \cdot k=\sum_{i=1}^d \omega_i k_i\) with \(k\in\mathbb{Z}^d\backslash\{0\}\) an integer vector, which usually are related to Fourier modes associated to the perturbing function and where the frequency vector \(\omega\) often depends upon the slow (action) variables. Such expressions appear in the denominator of (formal) Fourier expansions of the object one aims to construct (e.g., a generating function or the formal expansion of a quasi-periodic solution). Since \(\omega\cdot k\) may became arbitrarily small for any vector \(\omega\in\mathbb{R}^d\) as \(k\) varies, the convergence of the perturbative series is in doubt.
Kolmogorov's scheme
Two main ideas are needed to overcome the convergence problems related to the appearance of small divisors: (i) keep the frequency of the motion fixed; (ii) use a Newton quadratic method (the name comes form the elementary tangent Newton method for finding roots of real functions) to control the growth of the remainder terms. More specifically, Kolmogorov (Kolmogorov, 1954) constructed a (real-analytic), near-identity, symplectic transformation \(\phi_1\) transforming a Hamiltonian of the form \(H=K+\epsilon P\ ,\) with \(K\) a non-degenerate Kolmogorov normal form as in (3), into a new Hamiltonian of the form \[\tag{4} H_1:=H\circ\phi_1=K_1+\epsilon^2 P_1, \]
with \(K_1\) again in non-degenerate Kolmogorov normal form with the same frequency vector \(\omega\) as \(K\ .\) Once this is achieved, one can iterate the construction to obtain a sequence of symplectic transformations \(\phi_j\) so that \[ H_j:=H\circ\phi_1\circ\cdots\circ\phi_j=K_j+\epsilon^{2^j} P_j \] with \(K_j\) a non-degenerate Kolmogorov normal form with fixed frequency vector, and \(P_j\) a real-analytic perturbation. Indeed, the equations leading to the determination of the symplectic transformation \(\phi:(y',x')\to(y,x)\) may be (essentially uniquely) solved and admit as generating function a (real-analytic) function of the form \[ g(y',x)=y'\cdot x+\epsilon \Big(b\cdot x+s(x)+ y'\cdot a(x)\Big) \] where, \(b\) is a constant vector, while \(s\) and \(a\) are, respectively, a scalar and a vector-valued multi-periodic functions with vanishing average over \(\mathbb{T}^n\ .\) In the denominators of the Fourier expansion of \(s\) and \(a\) (and in the determination of the constant \(b\)) there appear the small divisors \(\omega\cdot k\ ,\) which are controlled through the Diophantine inequality (2). The super exponential decrease of \(\epsilon^{2^j}\ ,\) for small \(\epsilon\ ,\) allows to beat the growth of the norm (due to the small divisors) of the new perturbing functions \(P_j\ :\) in the limit as \(j\to\infty\ ,\) \(\phi_1\circ\cdots\circ\phi_j\) converges to a real-analytic symplectic transformation \(\phi_\epsilon\ ,\) \(H_j\to K_\epsilon\ ,\) with \(K_\epsilon=\lim_{j\to\infty} K_j=H\circ\phi_\epsilon\) a real-analytic non-degenerate Kolmogorov normal form with frequency \(\omega\ .\)
Arnold's scheme
Arnold (who was the first to provide a detailed proof of Kolmogorov's Theorem) followed a different approach (Arnold, 1963a), which, however, shared with Kolmogorov's scheme the two main ingredients. Arnold considered a nearly-integrable Hamiltonian system of the form \(H:=K(y) + \epsilon P(y,x)\) real analytic in a complex neighborhood \(D_0\) of \(\{y_0\}\times\mathbb{T}^n\ ,\) where \(y_0\) is such that \(\partial_y K(y_0)=\omega\) is Diophantine and \(\partial^2_y K\) is invertible on \(D_0\ .\) One then constructs a near-identity symplectic transformation \(\phi_1: D_1\to D_0\) transforming \(H\) as in (4) with \(K_1=K_1(y')\) (i.e., integrable). The new domain \(D_1\) is a complex neighborhood of \(\{y_1\}\times\mathbb{T}^n\) contained in \(D_0\ ,\) and with the property that \(\partial_{y'} K_1(y_1)=\omega\) (same frequency) and \(\partial^2_{y'} K_1\) is invertible on \(D_1\ .\) This is not difficult to achieve, by classical averaging theory, through a symplectic transformation associated to a near-identity generating function \(g(y',x)=y'\cdot x + \epsilon \tilde g(y',x)\ ,\) with \(\tilde g\) a trigonometric polynomial in \(x\) having degree \(\delta\) depending on \(\epsilon\) (\(\delta\) can be chosen as \((\log \epsilon^{-1})^p\) and being related to a cut-off of the high Fourier modes of the perturbation). The iteration leads to a sequence of Hamiltonians \(H_j=K_j+\epsilon^{2^j} P_j\) closer and closer to integrable but in shrinking domains \(D_j\ .\) In the limit the projection onto the action variables of \(D_j\) is a single point \(y_*\ .\) Nevertheless, one can show that, pulling back the dynamics to \(\{y_*\}\times \mathbb{T}^n\ ,\) there corresponds a Diophantine KAM torus for the original Hamiltonian \(H\ .\)
Moser's differentiable case
In dealing with a finitely differentiable perturbation \(P\) there appears an extra technical problem. Namely, due to the presence of the small divisors, during the iteration scheme one loses derivatives at each step. Moser (inspired by the famous work by J. Nash on the \(C^\infty\) imbedding of Riemannian manifolds) introduced a smoothing technique (via convolutions), which re-stores at each step of the Newton iteration a certain number of derivatives. The super-exponential convergence of the Newton scheme is fast enough to compensate also for the smoothing leading to a convergent algorithm. Later, Moser developed different and sharper methods, using, e.g., a characterization of differentiable functions through approximations by real-analytic ones in smaller and smaller complex neighborhoods of real domains. Thus, by a quantitative approximation of differentiable functions by means of real-analytic functions, one can construct for the analytic approximations real-analytic, invariant tori; such approximate solutions are analytic in shrinking domains and in the limit converge to differentiable solutions of the original problem.
The analytical tools needed in KAM proofs are classical and involve, in particular:
- exponential decay of Fourier coefficients of analytic functions
- quantitative versions of the classical implicit function theorem in real-analytic settings
- Cauchy estimates, which allow to bound the sup-norm of derivatives of analytic functions in smaller domains in term of the sup-norm of the function divided by the loss of the extension of the domain
- quantitative analysis of the PDE \(\sum_{j=1}^d \omega_j \partial_{x_j} u= f\) where \(f\) is real-analytic function on \(\mathbb{T}^d\) with vanishing average and \((\omega_1,...,\omega_d)\) a Diophantine vector.
- In a nearly-integrable analytic Hamiltonian system with \(2n\) degrees of freedom, the Kolmogorov set, i.e., the union of the persistent KAM tori, fills locally a region in phase space of density \(1-O(\sqrt{\epsilon})\ ,\) as \(\epsilon\) goes to zero. While the dynamics on the Kolmogorov set trivializes (being conjugated to a linear quasi-periodic translation on \(\mathbb{T}^n\) with a Diophantine frequency vector), in its complement (which asymptotically represents a small region of measure \(O(\sqrt{\epsilon})\)) the dynamics can be very complicated, exhibiting, in many cases, "random motions" or "Arnold diffusion".
- In nearly-integrable Hamiltonian systems, Kolmogorov's non-degeneracy condition is equivalent to require that \(\det \partial_y^2 K(y_0)\neq 0\ ,\) which, in turn, means that the frequency map \(y\to \omega(y):=\partial_y K(y)\) is a local diffeomorphism in a neighborhood of \(y_0\ .\)
- The global geometry of the Kolmogorov set is simple: the fibers of the set (i.e., the individual KAM tori) are level sets of a global \(C^\infty\) symplectic map \(\phi_*(\eta,x)\) as the \(n\) vector \(\eta\) varies in a Cantor-like \(n\)-disk of almost full density. This phenomenon may be interpreted by saying that nearly-integrable Hamiltonian systems are integrable over Cantor sets (Pöschel, 1982, Chierchia and Gallavotti, 1982).
- The Kolmogorov symplectic map \(\phi_\epsilon\) and the Kolmogorov normal form \(K_\epsilon\) (see above) depend analytically upon the perturbative parameter \(\epsilon\ .\) Therefore quasi-periodic trajectories taking place on KAM tori admit a convergent series expansions in \(\epsilon\). This fact, which was first observed by Moser (1967), solves a long-standing problem about the convergence of Lindstedt series (i.e., \(\epsilon\)-power series expansions of formal quasi-periodic solution with Diophantine frequencies). Direct proofs, based upon delicate and lengthy combinatorial arguments, of the convergence of Lindstedt series (i.e., proofs avoiding KAM fast iteration methods) were found in the late 1980's (H. Eliasson) and early 1990's (G. Gallavotti, L. Chierchia and C. Falcolini).
Applications and extensions
Iso-energetic tori and perpetual stability
The tori found through Kolmogorov's (or Arnold's) scheme have, as \(\epsilon\) varies, the same frequencies, but different energies. Arnold noticed that, instead, one could keep fixed the ratios of the frequencies and the energy so as to analytically continue KAM tori on a fixed energy surface. The analytical non-degeneracy condition to achieve this (in the nearly-integrable setting) is \[ \det \left(\begin{matrix}\partial^2_y K & \partial_y K \\ \partial_y K & 0\end{matrix}\right) \neq 0 \] (this is a \((n+1)\times(n+1)\) matrix having as last column and as last row the gradient of \(K\) and a 0).
Iso-energetic non-degeneracy leads, in low dimensional nearly-integrable systems, to perpetual stability: an energy level for a system with two degrees of freedom is a 3-dimensional surface and, for small perturbation, an iso-energetically non-degenerate, nearly-integrable systems admits a positive measure set of invariant two dimensional tori (which are graphs over the angle variables); thus such tori separate the energy level, and a generic trajectory either lies on an invariant torus or is trapped among two of them. In both cases no escape is possible, and the action variables stay forever close to their initial values ("perpetual stability").
Properly degenerate KAM theory
One of the original motivations for KAM theory was to find bounded motions in the planetary many body problem (i.e., a mechanical system formed by \(1+N\) point-masses, one of which is much larger than the other, interacting only through gravity). It is a classical fact that such a system may be seen as a perturbation of \(N\) decoupled two-body systems (star-planet). However, the limiting unperturbed Hamiltonian is highly degenerate, since it does not depend on the full set of action variables (proper degeneracy). In general, perturbations of properly degenerate Hamiltonian systems may admit no KAM tori. However, under suitable assumptions on the (average over the fast angles of the) perturbation KAM tori do exist:
Theorem (Arnold, 1963b) Let \((y,x)\in \mathbb{R}^n\times\mathbb{T}^n\) and \((p,q)\in\mathbb{R}^{2m}\) be couples of conjugate symplectic variables and let the Hamiltonian \(H=K(y)+\epsilon P(y,x,p,q)\) be real-analytic in a neighborhood of \(\{y_0\}\times\mathbb{T}^n\times\{0,0\}\ .\) Denote by \(\bar P\) the secular perturbation (i.e., the average over the fast angles \(x\) of \(P\)) \(\bar P(y,p,q)=\int_{\mathbb{T}^n} P(y,x,p,q) dx/(2\pi)^n\) and by \(r=(r_1,...,r_m)\) the vector with components \(r_i=(p_i^2+q_i^2)/2\) (for \(i=1,...,m\)). Assume that \(\det \partial_y^2 K(y_0)\neq 0\ .\) Assume also that the secular perturbation has an elliptic equilibrium\[ \bar P= \bar P_0(y)+ \Omega(y)\cdot r + \frac12 A(y) r\cdot r + O(|r|^3)\ ,\] with \(\det A(y_0)\neq 0\ .\) Then, if \(\epsilon\) is small enough, in a neighborhood of \(\{y_0\}\times\mathbb{T}^n\times\{0,0\}\) there exists a positive measure set of initial data whose evolution lies on \((n+m)\)-dimensional tori close to \(\{y_0\}\times\mathbb{T}^n\times\{r_k=\epsilon^a, \ \forall\ k\}\) for a suitable \(a>0\ .\)
This theorem, or refinements of it, is at the basis of the application of KAM theory to the planetary many body problem; a complete proof of such result, however, was published only in 2004 and is due to M. Herman and J. Fejóz.
Weaker non-degeneracies
To extend the validity of KAM theory it is important to weaken the non-degeneracy conditions. As mentioned above, Kolmogorov's non-degeneracy condition for nearly-integrable systems with Hamiltonian \(H_\epsilon=K(y)+\epsilon P(y,x)\) means that the frequency map \(\omega=\partial_y K\) is a local diffeomorphism. Rüssmann pointed out (Rüssmann, 1989) that it is sufficient (and in a suitable sense also necessary) to assume that the image of the frequency map \(y\to \omega(y)\) does not lie in any hyperplane, (more precisely, for a ball \(B\ ,\) \(\omega(B)\) does not lie in any hyperplane passing through the origin). A similar condition (that suites better differentiable settings) due to Arnold and Pyartli is to require that the frequency map \(\omega\) is skew at some point \(y_0\ .\) This means that there exists a smooth curve \(t\in(-1,1)\to u(t)\in\mathbb{R}^n\) passing through \(y_0=u(0)\) such that, if \(\alpha(t)\) denotes the lifted curve \(\omega\circ u(t)\ ,\) then the matrix \([\alpha(0),\alpha'(0),...,\alpha^{(n-1)}(0)]\) is invertible. Under these types of non-degeneracy conditions one can guarantee that, under small enough perturbations, there exists a positive measure set of initial data evolving on maximal KAM tori for \(H_\epsilon\ .\)
Lower dimensional tori
Orbits of great interest for KAM theory are also quasi-periodic trajectories spanning lower dimensional tori, i.e., orbits \(z(t)=\phi_H^t(z_0)\) such that the closure of the set \(\{z(t): t\in\mathbb{R}\}\) is diffeomorphic to \(\mathbb{T}^d\) with \(1< d<n\ ,\) \(n\) being the number of degrees of freedom (i.e., half of the dimension of the phase space). At variance with maximal KAM tori, the union of lower dimensional tori form a set of Lebesgue measure zero in phase space; nevertheless they are important in order to understand the dynamics and for extensions of KAM theory to PDEs. To fix ideas, consider the normal form of a lower dimensional elliptic torus \[\tag{5} K(y,x,p,q;\xi):=E(\xi) + \omega(\xi)\cdot y + \frac12 \sum_{j=1}^m \Omega_j(\xi)(p_j^2+q_j^2)\]
where \((y,x)\in\mathbb{R}^d\times \mathbb{T}^d\) are (partial) action-angle variables; \((p,q)\in\mathbb{R}^{2m}\) are conjugated variables; \(\Omega_j(\xi)>0\) and \(\xi\) is a real \(d\)-dimensional parameter (for example, \(\xi\) might be a fixed action \(y_0\) around which one is making a Taylor expansion). The set \(\mathcal{T}^d_0:=\{y=0\}\times\mathbb{T}^d\times\{p=0=q\}\) is an invariant \(d\)-dimensional torus for \(\phi_K^t\ :\) \(\phi_K^t(0,x_0,0,0)=(0,x_0+\omega(\xi)t, 0,0)\ .\) Such a torus is linearly stable (elliptic), and the dynamics close to it, in the \((p,q)\)-variables is just given by harmonic oscillations with frequencies \(\Omega_j(\xi)\) (tangential frequencies). Under suitable regularity and non-degeneracy assumptions (on the inner and tangential frequencies) such tori are persistent.
For example, let \(\xi\) vary in a closed set \(\Pi\) of positive \(d\)-dimensional Lebesgue measure; let \(\xi\to\omega(\xi)\) be a Lipschitz homeomorphism and let \(K\) and \(P(y,x,p,q;\xi)\) be real-analytic in the symplectic variables \((y,x,p,q)\) and Lipschitz continuous in \(\xi\ .\) Assume that \(\Omega_j(\xi)\neq\Omega_i(\xi)>0\) for all \(i\neq j\) and \(\xi\in \Pi\ .\) Assume also the following (Melnikov -Pöschel) condition \[{\rm meas}\,\Big(\{\xi\in\Pi: \omega(\xi)\cdot k + \Omega(\xi)\cdot \ell=0\}\Big)=0 \ ,\quad \forall\ k\in\mathbb{Z}^d\backslash\{0\}\ ,\forall\ \ell\in\mathbb{Z}^m\ {\rm with}\ \sum_{j=1}^m|\ell_j|\le 2\ . \] Then, there exists \(\epsilon_*>0\) and a Cantor set \(\Pi_*\subset\Pi\) of positive measure such that to each \(\xi\in\Pi_*\) there corresponds, for any \(0<\epsilon<\epsilon_*\ ,\) a torus \(\mathcal{T}_\epsilon^d(\xi)\) invariant for \(\phi_{K+\epsilon P}^t\ .\)
We remark that this kind of result admits many generalizations, which are particularly important for infinite dimensional extensions. The partially hyperbolic case, whose normal form is give by (5) with \((p_j^2+q_j^2)\) replaced by \((p_j^2-q_j^2)\) is much simpler (as in this case the tangential frequencies do not resonate with the inner ones); see (Graff, 1974).
Hamiltonian PDE's
KAM theory can be partially extended to infinite dimension, i.e., to partial differential equations (PDEs) carrying a Hamiltonian structure. Examples of such equations are: the wave equation, the (stationary) Schrödinger equation, KdV, etc. Under suitable hypotheses, nonlinear perturbations of these equations may be reduced to infinitely many coupled dynamical (ordinary differential) equations (e.g., for the wave equation one obtains infinitely many coupled harmonic oscillators). It is then possible to find quasi-periodic solutions corresponding to the embedding of a linear quasi-periodic flow on a finite dimensional torus into the infinite dimensional phase space associated to the equation. Also almost-periodic motions have been considered (i.e., trajectories with infinitely many independent frequencies). Several results in these directions have been obtained starting from the 1990's; see (Kuksin, 2004).
- Arnold , V I (1963a). Proof of a Theorem by A. N. Kolmogorov on the invariance of quasi-periodic motions under small perturbations of the Hamiltonian. Russian Math. Survey 18 : 13-40.
- Arnold , V I (1963b). Small divisor problems in classical and Celestial Mechanics. Russian Math. Survey 18 : 85-191.
- Arnold , V I (1964). Instability of dynamical systems with many degrees of freedom. Dokl. Akad. Nauk SSSR 156 : 9-12.
- Chierchia(1982). Smooth prime integrals for quasi-integrable Hamiltonian systems Il Nuovo Cimento. B. Serie 11 67: 277-295.
- Féjoz , J (2004). Dèmonstration du `théorème d'Arnol'd' sur la stabilité du système planétaire (d'après Herman). Ergodic Theory Dynam. Systems 5 : 1521-1582.
- Graff , S (1974). On the continuation of stable invariant tori for Hamiltonian systems. J. Differential Equations 15 : 1-69.
- Herman, M-R (1983). Sur les courbes invariantes par les difféomorphismes de l'anneau. Vol. 1. Astérisque 103: i+221.
- Kolmogorov, A N (1954). On the conservation of conditionally periodic motions under small perturbation of the Hamiltonian. Dokl. Akad. Nauk. SSR 98: 527-530.
- Kuksin, S B (2004). Fifteen years of KAM for PDE. Geometry, topology, and mathematical physics, Amer. Math. Soc. Transl. Ser. 2 212: 237-258.
- Melnikov , V K (1965). On certain cases of conservation of almost periodic motions with a small change of the Hamiltonian function. Dokl. Akad. Nauk SSSR 165: 1245-1248.
- Moser , J K (1962). On invariant curves of area-preserving mappings of an annulus. Nach. Akad. Wiss. Göttingen, Math. Phys. Kl. II 1 : 1-20.
- Moser , J K (1967). Convergent series expansions for quasi-periodic motions. Math. Ann. 169 : 136-176.
- Pöschel , J (1982). Integrability of Hamiltonian systems on Cantor sets. Comm. Pure Appl. Math. 35 : 653-695.
- Rüssmann , H (1970 ). Kleine Nenner. I. Über invariante Kurven differenzierbarer Abbildungen eines Kreisringes. Nach. Akad. Wiss. Göttingen, Math. Phys. Kl. II 1970: 67-105.
- Rüssmann , H (1989). Nondegeneracy in the perturbation theory of integrable dynamical systems. Number theory and dynamical systems (York, 1987), London Math. Soc. Lecture Note Ser. 134 : 5-18.
Internal references
- Paul M.B. Vitanyi (2007) Andrey Nikolaevich Kolmogorov. Scholarpedia, 2(2):2798.
- Jan A. Sanders (2006) Averaging. Scholarpedia, 1(11):1760.
- Sylvio Ferraz-Mello (2009) Celestial mechanics. Scholarpedia, 4(1):4416.
- Alessandra Celletti (2008) Computational celestial mechanics. Scholarpedia, 3(9):4079.
- James Meiss (2007) Dynamical systems. Scholarpedia, 2(2):1629.
- Eugene M. Izhikevich (2007) Equilibrium. Scholarpedia, 2(10):2014.
- Ferdinand Verhulst (2007) Hamiltonian normal forms. Scholarpedia, 2(8):2101.
- James Meiss (2007) Hamiltonian systems. Scholarpedia, 2(8):1943.
- James Murdock (2006) Normal forms. Scholarpedia, 1(10):1902.
- Andrei D. Polyanin, William E. Schiesser, Alexei I. Zhurov (2008) Partial differential equation. Scholarpedia, 3(10):4605.
- Jeff Moehlis, Kresimir Josic, Eric T. Shea-Brown (2006) Periodic orbit. Scholarpedia, 1(7):1358.
- Anatoly M. Samoilenko (2007) Quasiperiodic oscillations. Scholarpedia, 2(5):1783.
- Philip Holmes and Eric T. Shea-Brown (2006) Stability. Scholarpedia, 1(10):1838.
- David H. Terman and Eugene M. Izhikevich (2008) State space. Scholarpedia, 3(3):1924.
- Christophe Golé (2010) Symplectic maps. Scholarpedia, 5(3):3722.
- Alain Chenciner (2007) Three body problem. Scholarpedia, 2(10):2111.
Recommended reading
- Arnol'd, V I; Kozlov, V V and Neishtadt, A I (2006). Mathematical Aspects of Classical and Celestial Mechanics, Dynamical Systems III Series: Encyclopaedia of Mathematical Sciences. Springer-Verlag 3rd ed. Vol. 3: xiv+518.
- Moser, J K (1966). A rapidly convergent iteration method and non-linear partial differential equations Ann. Scuola Norm. Sup. Pisa 20: 499-535.
- Moser, J K (1973). Stable and random motions in dynamical systems. With special emphasis on celestial mechanics Hermann Weyl Lectures, the Institute for Advanced Study, Princeton, N. J. Annals of Mathematics Studies 77: viii+198.
See also
Averaging, Aubry-Mather theory, Chaos, Computational celestial mechanics, Dynamical Systems, Hamiltonian Dynamics, Hamiltonian Normal Forms, N-Body Simulations, Normal Forms, Standard map, Symplectic maps, Three body problem

subset: commoncrawl


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Wizz Air is opening a new base in the UK - with flights to Spain, Greece, Italy and Malta
Wizz Air has announced it is opening a new base at Gatwick Airport, with flights available to some of the most popular European holiday destinations.
Gatwick will become the third UK base for the budget airline, which also operates from London Luton and Doncaster in Sheffield.
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Wizz Air will launch four routes from its new base to destinations in Spain, Greece, Italy and Malta.
The airline plans to allocate one Airbus A321 aircraft to Gatwick, which has recently seen Virgin Atlantic, British Airways and Easyjet all vacate in the wake of a financially difficult year for the travel industry due to the coronavirus pandemic.
Wizz Air flights will begin running to Athens, Naples, Malta and Lanzarote from 22 October, with fares for the new routes starting at £21.99 each way. Tickets for the new flight routes are now available to book online.
Travellers can also use the airline's Flex service as an add-on to their fare, which will allow flights to be cancelled up to three hours before departure, with 100 per cent of the fare immediately reimbursed in airline credit.Customers can change the dates and routes of their flights an unlimited number of times.
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While Wizz Air has already been flying from Gatwick since 2016, the new base marks the first time it will be permanently set up at the London airport.
The announcement follows the airline's recent expansion at Doncaster Sheffield, which was confirmed just a week ago, on 13 August. Its new permanent home at the Yorkshire-based airport will serve seven new routes, including Alicante and Malaga in Spain, Larnaca in Cyprus, Faro in Portugal, Lublin in Poland, Kosice in Slovakia and Suceava in Romania.
These routes will all operate from 22 October, with tickets priced from £17.99.
In addition to its three UK bases, Wizz Air also offers flights from 11 airports in the UK.


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After nearly two months of tinder-dry weather, the B.C. communities hit hardest by wildfires could be seeing a bit of rain this weekend.
Environment Canada is forecasting a chance of scattered showers along the coast and into the southern Interior and Cariboo.
But BC Wildfire Service chief information officer Kevin Skrepnek said exactly how the wet weather will affect the 126 active fires in the region won't be known until the rain arrives.
The rain is forecast to come with wind, which could be problematic for crews battling more volatile fires – such as Elephant Hill burning at 117,000 hectares in size near Clinton and the Chilcotin fire complex.
"Rain is a real wild card to try and forecast this far out," Skrepnek said Wednesday.
Then there's the chance of the upcoming rain turning to thunderstorms with lightning, which has been the culprit behind about 70 per cent of wildfires in 2017.
"Whether [the rain's] a help or a hindrance remains to be seen," he said. "Certainly though, getting some rain on these fires would be welcome but if we're getting this unstable weather with it it might be a bit of a mixed blessing."
Weather forecasts suggest the rain will begin Saturday on the coast, with scattered showers in areas of the Cariboo and southern Interior through to Sunday, when weather is expected to remain at cooler temperatures.
Until then, back country areas remained off-limits to the public and off-roading bans also remain in effect.
Backcountry recreation sites still closed due to #BCwildfires - daily assessments underway to determine re-opening https://t.co/2GSBr2OANk
— Ashley Wadhwani (@ashwadhwani) August 9, 2017
How much rain is enough rain?
In July, a day of rain in Williams Lake had offered a glimmer of hope for the then-evacuated city.
RELATED: It's raining in Williams Lake, but not for long
The next day the BC Wildfire Service reported 32 more fires were caused by lightning in the Cariboo and Kamloops fire regions.
At the time, Environment Canada meteorologist Cindy Yu said short spurts of rainfall allows for a break in the heat but aren't enough to make a real difference on larger wildfires burning.


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Moving into a new home is a stressful but rewarding experience. This customer had just recently moved into a property in the neighbouring and historic Flintshire town of Mold, and was keen to make some renovations.
High on the agenda was the restoration of a large number of Ceramic floor tiles spread throughout multiple downstairs rooms. Although undecided as to whether to keep the tiles for the long-term or have them removed, the customer did resolve to have them professionally cleaned, especially since the previous owner had left the grout in a miserable state.
Upon initial inspection I noticed that the previous owner had attempted to put more grout on top of the existing one in some areas, the problem with this being that the two groups were of different colours. This left the grout looking inconsistent and dirtier than it actually was. I mentioned to the customer that I could re-colour the grout for them after cleaning the floor and that it would be the best solution in this situation.
Cleaning and Colouring Ceramic Tiled Floor Grout
As mentioned, the tiles themselves needed a deep clean, and so I set about applying a strong dilution of Tile Doctor Pro Clean liberally over the floor and leaving it to dwell for a short period. This allows the product, a high alkaline cleaner suitable for use on most natural stone floors, to seep into the pores of the stone to lift out and dissolve ingrained dirt and stains. Once I had allowed it to dwell I scrubbed the product into the tiles using a black pad fitted to a rotary machine.
Of course I also paid special attention to the grout lines, carefully cleaning them using more Pro Clean. With some floors I usually give the floor an acid wash at this with Tile Doctor Grout Clean Up to remove grout smears and mineral salt deposits but this was not necessary on this particular occasion.
Once I had cleaned the entire floor I gave it a thorough rinse with water which was then extracted using a wet vacuum before leaving the floor to dry for two hours assisted with an air mover to speed up the process. When happy that the tiles were completely dry and ready to be re-coloured I went ahead and carefully applied a new black Grout Colourant that matched very well with the natural colour of the stone, giving the whole floor a smarter appearance.
The customer was really pleased with the result and left some nice feedback on the job as a whole:
"Excellent experience all round, from initial consultation through to colour advice and scheduling work. Cannot speak highly enough of Joe, thoroughly professional and an amazing result. Highly recommended."
Source: Ceramic Tile and Stone Restoration Service in Shropshire


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Emily and Virginia book by Oregon Book Award Finalist Robert McDowell, a poet, performer, storyteller, public speaker, narrative healer, educator, coach, mentor, editor, and author of 16 books, including the bestselling Poetry As Spiritual Practice.
Sign up below to receive the full-length video of Robert's engaging dialogue with Dr. Rama Mani and sign up for his e-newsletter.
Reach Robert McDowell via our contact form or email him directly.
"A well-guided trip opens new vistas, which you did for me. The session with you gave me perspectives and approaches and provided a rhythm to unknown potentials and made them accessible."
"An email just landed in my inbox that intrigued me, and I clicked through to the person's web page. And fell in love with the man's description of himself, who he is, and what he's here to do. I felt so invited, inspired, served. May we ALL represent ourselves so beautifully!"
"Robert McDowell prepared an exceptionally interesting class on Emily Dickinson: The Divine Feminine Prophet.… Robert is a fine teacher, poet, and philosopher who has put together a program that offered, to borrow a line from Emily Dickinson: 'that sublimer lesson, a faith in things unseen, and in a life again, nobler, and much more blessed.'"
"I always remember how special it was to be in your class. I still feel the influence your class had on me—my worldview, my appreciation of others and their feelings and opinions.… Thank you for what you did for me then and for what you do now for the people you work with—your talent and teaching were a gift to me."
"Robert McDowell is the very spirit of poetry and sumptuous prose. He is a wordsmith and a soul crafter of the highest order and his talks and seminars are veritable temples of magic. If ever he is in your community for a talk or workshop, go!"
"I just finished listening to your teleclass, and I was deeply moved by your talk. Thank you so much for partnering with us. Your work is a blessing to the world."
"My heart is so very full after having the blessed opportunity of attending your teleseminar this evening. How comforting to know that men of your amplitude and sensitivity are cognizant and brimming with the magical means to help pave the way for a more intimate and authentic sacred connection."
"I love the way you put words together, and will indeed, look forward to more of your inspiration, through readings and future workshops. You are so talented, and as I see it, reaching out, is one of your greatest qualities."
"Robert McDowell's online workshop on Emily Dickinson was a thrilling and profound experience. He avoided all superficialities and cut straight and deep to the chase with Dickinson's poems, letters, his own concise and brilliant erudition, and input from many relevant figures.… A beautifully planned, shaped, and conducted learning experience."
"Poetry exposes me to a different way of experiencing the world. When I read Poetry as Spiritual Practice, I instantly translated the poems into pictures. I can see fields of grain or rain in Autumn. It is fascinating to see all the patterns and rhymes that can be woven into language."
"Bless you, Robert. I have enjoyed your verse, the Reaper Essays, and find your spiritual practice model so acute and helpful. Despite Auden's injunction, art can make things happen. Getting into the poesis mindset is one of the healthiest practices a human can do."


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Abortion Pill Near Columbus, Montana
Mifepristone, also referred to as RU-486 or the abortion pill, is the drug used in the pharmaceutical abortion method. Based on the ultrasound done in our facility, your pregnancy must be between 6 and 11 weeks to qualify for this option. The time since your last regular menstrual cycle usually matches with the ultrasound. If the pregnancy continues, several of the medications used in the abortion pill results in severe birth abnormalities. You could require extra medication or a procedural abortion (aspiration) to stop the pregnancy in the unlikely event that the abortion pill does not work. Visit our office today to discuss your options in more depth.
Two distinct drugs are used in an abortion pill near Columbus, MT. Mifepristone is the name of the first drug. A medication called mifepristone inhibits the hormone that promotes pregnancy growth. This is administered in addition to Misoprostol, another drug. Your body flushes out the pregnancy thanks to misoprostol. This two-step procedure is a secure and efficient way to get an abortion at home or any other location you choose. You must schedule a follow-up appointment 10–14 days after you take the abortion pill.
Bleeding, cramps and nausea are frequent side effects of taking the abortion pill. Usually, these signs appear after taking misoprostol (2nd step of the medication abortion process). From moderate to excessive bleeding, it might vary from person to person. Many folks claim that this feels like a very burdensome period. Cramping may also occur during this process. As directed for your abortion pill, you may take painkillers. In general, you can relieve cramping during this abortion pill treatment by doing whatever you do to relieve your regular period cramps.
How Effective Is The Abortion Pill near Columbus, Montana
Studies have demonstrated the great effectiveness of Mifeprex and misoprostol for those individuals who have taken the abortion pill near Columbus, Montana. Millions of women worldwide have used mifepristone with success. Approximately 7 out of every 100 women who use this procedure will not successfully end the pregnancy. For some ladies, a surgical abortion, instead of the abortion pill, can be performed through our physicians. We will provide you with the care that you need. Whether it may be an abortion pill near Columbus, MT or a surgical abortion, the physicians at Wellspring Health Access are here for you. Call today to find out more.


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Keep your grill going strong this fall with these 3 helpful tips
Tips for Fall Grilling
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Find out what's new from Weber, including recipes, product information, personalized communications and so much more!
As I watch leaves start to fall off the trees here in southwest Ohio, I'm having a hard time coming to grips with the arrival of fall.
While I grill year round (don't we all?!), there is a lot to be said for the joys of hot summer nights at the kettle. Oh, and to all of my friends in perennial warm climates, or in the southern hemisphere just now planning for summer, yes, I'm jealous.
Grilling in cold and darker weather is not insurmountable, it just involves more clothing and better planning. Here are a few things to keep in mind as the seasons change.
If your grill goes to rest over the winter months, a good clean come spring will suffice. However, if winter is simply the halfway point of your year long grill adventure, a fall clean is probably a good idea. My last thorough clean was in May. As I hoisted the lid on my Summit Grill Center last night, I peered down into the grates and drip pan. It needs a good clean now, as well as the removal of some fallen spears of asparagus.
This is the time of year I wait for an unusually warm day to "fall clean" my grills. It's a lot easier to do when the weather is still nice. Cleaning in the fall is just as important as spring cleaning.
Winter means snow and wind and as of last winter, more snow. For a change of scenery, I tend to move the grills around our back deck. However, as winter approaches, I look to put them in a more sheltered and protected location. I want to minimize the impact from falling snowflakes and stiff breezes, as both can impact cooking times and also, as equally important, griller comfort.
Grill covers are beyond valuable in the cold weather months. Just pull the cover off, no need to brush the snow off or de-ice your grill. Our covers have a 3 year warranty and are built to withstand the cold weather months.
While the cold weather is one thing, winter also means early sunsets. It gets dark early, real early. In the summertime, we rarely think about it, but as the days grow shorter, make sure you have adequate outside lighting to illuminate the grills grates, or better yet, the entire grill area. Check bulbs and switches to make sure everything is in working order. It's a lot easier to do it now than during the first snowfall. I highly recommend getting your grill set up with a grill handle light. You will be thankful when you can see the food on your grill!
It's a little more of a challenge to grill in the fall and winter, but with a little planning and effort, it makes the adventure easier and more enjoyable...well, sorta enjoyable. It's still cold outside.


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EG Pic Quiz
About EG Pic Quiz Games
The aim of the game is to guess what's on the picture. While using the least number of clicks.
Each click opens up a portion of the photo. The earlier the answer becomes clear, the larger the bonus will be.
Thanks to this game you'll easily increase your vocabulary, guessing skills and attentiveness.
It's a new quiz in the casual puzzle genre.
Categories and tags of the game : Arcade, Ecapsgames, Ecapsgames.com, Egpicquiz, Egquizgames, Puzzle


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By Rebecca Solnit
Mark Lombardi's art consists of colossal drawings of networks of power, connecting politicians, capitalists, and corporations into intricate maps, like medieval cosmology or kabbalah diagrams, whose huge arcs and circles linking the small handwritten names are as visually beautiful as they are politically daunting. His most famous work was about the BCCI (Bank of Credit and Commerce International, also known as the Bank of Crooks and Criminals) banking scandal. It linked up the Bin Laden and Bush families long before Fahrenheit 9/11, even before the 2000 election and Bush's illegitimate apotheosis as president.
New York critic Frances Richard wrote of this work:
"Lombardi's drawings — which map in elegantly visual terms the secret deals and suspect associations of financiers, politicians, corporations, and governments — dictate that the more densely lines ray out from a given node, the more deeply that figure is embroiled in the tale Lombardi tells…. The drawing is done on pale beige paper, in pencil. It follows a time-line, with dates arrayed across three horizontal tiers. These in turn support arcs denoting personal and corporate alliances, the whole comprising a skeletal resume of George W. Bush's career in the oil business. In other words, the drawing, like all Lombardi's work, is a post-Conceptual reinvention of history painting…."
After September 11, 2001, the FBI visited the Whitney Museum to examine his drawings for clues they might yield about the conspiracy that gave rise to the catastrophe.
Lombardi committed suicide in March of 2000, for complex reasons, but it's easy to imagine him as a character in a Jorge Luis Borges story dying of Borgesian reasons. For his drawings recall Borges's library of Babel, his Garden of Forking Paths, the Zohar, Zeno's paradox or the aphorism by Pascal Borges loved, "The universe is a sphere whose center is everywhere and circumference is nowhere." Borges's parables and stories are attempts to grasp the infinite complexity of the world, and his version of Lombardi would have died of despair of ever approximating the reach and intricacy of these networks.
Lombardi's work is often regarded as evidence of sinister conspiracies by people who assume that "they" are thus linked up but "we" are not. We are, actually, at least when we try to achieve anything political. Politics is networks, rhizomes, roots, webs, to use a few of the popular metaphors from the increasingly popular studies of complexity. A more cheerful Lombardi might have charted the links that connect Naomi Klein, the Argentina Horizontalidad populist movements against neoliberalism, the Zapatistas, the Yucatan campesinos who opposed the WTO in Cancun in 2003, the internationalistas who joined them, the US campus-based anti-sweatshop movement, the Sierra Club, Arundhati Roy, anti-Monsanto agriculturalists in India and Europe, on to Nigerian activists now shutting the operations of Chevron (based in San Francisco) and San Francisco activists against Bechtel Corporation (also based here), which links us back to Bolivian activists who beat Bechtel a few years ago. (Thanks to the Internet, speaking of networks, the global justice movement has been able to link causes and confrontations into an unprecedented meta-community able to act in concert internationally.)
In fact, right-wing think-tanks are probably lining up these affiliations and solidarities right now and portraying them as a conspiracy, as they have before. That's the rule of thumb: When we talk, it's a network; when they talk, it's a conspiracy. The sinister thing about Lombardi's BCCI drawing isn't that all these people, banks, and governments are linked up, but that they're linked up to screw you, me, and the world. That is to say, it's complexity that makes the drawing itself overwhelming, but intent that makes the denizens of the drawing scary.
Awakenings and Coincidences
One can imagine the characters of Adam Hochschild's wonderful new history, Bury the Chains, Prophets and Rebels in the Fight to Free an Empire's Slaves, as drawn by Lombardi, such is the complexity of the network Hochschild depicts while tracing the British antislavery movement from Quakers in London to slave rebellions in the Caribbean, from the 1780s when the movement began to the final, long-delayed abolition of slavery in the British Empire on August 1, 1838. The book is both a gripping history of a particular movement and a beautiful embodiment of the erratic, unlikely ways movements unfold — an unfolding that consists of multiple kinds of linkages. If Lombardi's is post-conceptualist history painting, Hochschild's book is likewise a kind of post-Great Man history writing, one with crowds, coincidences, and ocean currents looming up behind the key activists he delineates so beautifully.
One kind of linkage is coincidence. Another is friendship and the affinities of interests and emotions upon which friendships are based. Bury the Chains begins, in fact, with two remarkable series of coincidences that deliver up as their results two of the principal activists against slavery. Granville Sharp was the youngest of eight siblings who played music together and shared an evangelical piety. King George III thought he had the best voice in England. His brother, William Sharp, the king's physician, provided free medical care to the London poor. Jonathan Strong, a slave whose owner had pistol-whipped him viciously about the head and then threw him out on the street to die, came for treatment. Granville happened to be visiting that morning, and the brothers got Strong into a hospital; then, after his months of convalescence, they found him a job with a pharmacist. One day his owner encountered on the streets of London his former property healthy and fit, seized Strong, and sold him to a Jamaican plantation owner, arranging for him to be jailed until he could be shipped to the West Indies. The Sharp brothers intervened and managed to free him. "With this case," writes Hochschild, "the thirty-two-year-old Granville Sharp became by default the leading defender of blacks in London, and indeed one of the few people in all of England to speak out against slavery. And speak he would, vehemently, for nearly half a century. The fight against slavery quickly became his dominating passion."
Only one coincidence, the meeting with Strong, made Sharp an activist. But the string of events that brought the most pivotal activist into being was far stranger and more Lombardian. An antislavery activist, Olaudah Equiano, a former slave from what is now Nigeria via Barbados and Virginia whose autobiography later had a huge impact on the movement, saw a letter in the Morning Chronicle and London Advertiser on March 18, 1783. The letter recounted a case involving the British slave ship Zong. Equiano called on Sharp, and Sharp made the case a minor cause célèbre. It was an insurance case, on the face of it. The insurers challenged the claim of the Zong's captain that he had ordered 133 African captives thrown overboard alive in the mid-Atlantic because the ship's drinking water was running out. Jettisoning slaves insured as cargo would have led to compensation under those circumstances.
Human rights were never a consideration in the case. But the chief mate, afflicted with pangs of conscience, testified that there had been plenty of water. The murders took place to collect insurance on slaves who were sick and dying and therefore would not, on reaching land, become marketable commodities. The court found in favor of the captain and the ship's investors. Sharp then wrote indignant letters to several prominent clergymen, who mentioned the case in their sermons and writings.
The case of the Zong was far from over, and as the concerns it raised migrated onward through England, linkages began to build that would spark a potent anti-slavery movement. One Church of England clergyman who took up the case was Dr. Peter Peckard, who soon after became Vice-Chancellor of Cambridge University. When it was his turn to set the subject for the school's prestigious annual Latin composition prize, he chose Anne liceat invitos in servitutem dare? — "Is it lawful to make slaves of others against their will?" It was by no means a particularly likely choice. The Church of England's Society for the Propagation of the Gospel in Foreign Parts, governed in part by divinity professors from Oxford and Cambridge, derived significant revenues from Codrington, one of the biggest Barbados plantations. The place relied on branding, whipping, murdering, and constant terror to keep up the labor that worked the slaves to death. Slavery was outside the moral universe that even those propagating the gospel concerned themselves with, as Hochschild points out; the former slaver who wrote "Amazing Grace" worried about all sorts of minor sins long before he noticed that slavery might be a problem.
A scholarship student, Thomas Clarkson, won the 1785 Cambridge Latin Prize after devoting two months to researching and writing about slavery. But the winning mattered little, except that it drew attention to the essay and its writer, who would publish it in English as an antislavery tract. The publisher Clarkson found was a Quaker who introduced him to the few others, also Quakers, who not only believed slavery should be abolished but were willing to work for the great unlikelihood that someday it might be.
This chain of encounters and awakenings steered Clarkson away from a religious career into a passionate championing of the rights and humanity of the slaves in the British Empire. He quickly became the most effective activist the movement would have, one who gave the rest of his life — nearly half a century — over to the cause. Writing, investigating, talking, riding tens of thousands miles on horseback, he recruited, inspired, and connected the recruited and inspired into a movement. The Quakers who had organized a little earlier to abolish slavery had long needed a mainstream Anglican champion. In Clarkson they found a superb one, in close sympathy with them; he was by the end a Quaker in all but name.
Making a Movement
Some activists are born into their disposition and vocation, but many of the most passionate lead ordinary lives until some injustice or atrocity strikes them like lightning and they are reborn dedicated. Clarkson was such an activist, and he even had a transformative moment like Saint Paul on his way to Tarsus: riding to London, he got off his horse and sat down "disconsolate on the turf by the roadside and held my horse. Here a thought came into my mind, that if the contents of the Essay were true, it was time some person should see these calamities to their end."
With luck and dedication, he became that person. His movements and contacts among slave-ship doctors as well as theologians, Liverpool as well as London, amount themselves to a network whose complexity is comparable to some of Lombardi's diagrams. Like most high-profile activists, he had a committee behind him — nine Quakers, Granville Sharp, and another Anglican — who with him founded what was almost unprecedented then and common now, a nongovernmental organization, or as the shorthand for them goes, an NGO. (He also circulated the famous diagram of slaves packed into the hull of a ship that his Quaker colleagues published as a poster — a diagram still widely known and one of the great visual icons of inhumanity of all times.)
At various moments during the antislavery campaigns, there were widespread petitions to Parliament at a time when "petitioning" was one of the few rights available to ordinary citizens, sugar boycotts, since sugar was the principal West Indies slave product; local antislavery groups and sympathetic Parliamentarians; all the accoutrements, as Hochschild points out, of later human rights movements. The main sympathetic Parliamentarian was the wealthy, pious William Wilberforce, who opposed labor organizing and other extensions of rights and powers to the underclasses, but devoutly opposed slavery. (In the interim, he argued that whipping should not be abolished, but done only at night: some compromise is strategic, but some — like the Democrats looking for a nicer version of the war — is moral compromise.) Timid and conventional, he made an odd pairing with the radical, far-ranging Clarkson, but they remained friends for life.
Compromise ran all through this movement, or rather some of its members, while others were ardent revolutionists eager to see all the rights of man — and when women took over leadership in the 1820s, of women — granted. In this movement and any other, the utility of compromise is an arguable point (or one can argue instead for a kind of symbiosis of unbending activists and back-room-dealing ones, whereby the revolutionists extend the argument and make the reformers look reasonable — which is how the Sierra Club often looks at groups like Earth First! — and in due time, even revolutionists come to look reasonable, as did abolitionists once they had won).
For example, the movement long campaigned against the slave trade rather than the existence of slavery itself in the British Empire, on the grounds that it was a more winnable battle — and it was. (British sugar plantations were so energetically murderous that they required constant replenishment of the slave population from Africa, which is why it looked as though British slavery, unlike slavery in the United States, could be undone simply by closing down the maritime trade in human beings — i.e., the supply of fresh slaves.) The slave-trade struggle was won in 1807, while the abolition of slavery took more than another quarter century and even then limped forward with a six-year interim period when the slaves' labor was somehow to further compensate their masters, who had already been compensated in cash for loss of ownership of their fellow human beings. The most radical antislavery activist, Elizabeth Heyrick, had long before suggested that it was the slaves who were due compensation for their lives and labor. Still, the antislavery movement kept its eyes on the prize, clear that it was more important to free the slaves by any means necessary than to punish slavery's perpetrators.
Clarkson and his colleagues built a network consciously and conscientiously, recognizing that in doing so they were laying the foundations for the undoing of slavery. It stretched from the vast numbers of ordinary citizens who signed petitions and followed boycotts to the sympathetic witnesses who brought information back from Africa for what were, in essence, the first official human rights hearings in history, to the slaves themselves who turned up in London to testify or rose up in the Caribbean. (One of the things that distinguished the British abolition movement from the American was the fierce, effective slave revolts that terrified slaveholders and played a role on the road to abolition.) More fortuitous, or fortunate, or mysterious is the string of coincidences that brought the Zong to trial, the trial to Equiano's attention, Equiano into friendship with Sharp, Sharp to write to Peckard, Peckard to set the Latin prize topic as slavery, and Clarkson to be as inspired in his Latin as passionate in his conscience. It's one of those for-want-of-a-nail conundrums: how would it have come about had any element been absent?
Friendships and Atmospheres
Clarkson shows up on the periphery of other histories and other networks. In the 1790s, he moved to England's Lake District and became close friends with a poet who had also written a gold-medal-winning composition on slavery at Cambridge, this time an ode in Greek in 1792: the Romantic poet Samuel Taylor Coleridge. Clarkson's wife became Dorothy Wordsworth's close friend and key correspondent, and with this leap from the advancement of human rights to the advancement of British poetry, Hochschild bumps into a dizzyingly broad network of radical ideas stretching from the French Revolution to the vast slave revolution against the French in Haiti.
At that moment, some sense of what it means to be human was shifting, and the antislavery movement was part of that shift, as was the Romantic movement, with its cultivation of introspective awareness and its enthusiasm for liberation and revolution. Clarkson turns up briefly in Jonathan Holmes's Coleridge biography as someone who supported him in the depths of his opium addiction, while Wilberforce was close friends with Wordsworth's kindest uncle. Wordsworth himself wrote a sonnet to Toussaint l'Ouverture, who led the Haitian slave uprising, and William Blake and J. M. W. Turner both addressed slavery in their visual art (the moral if not the aesthetic ancestors of Lombardi). Others, such as the Wedgwood family, link the antislavery and poetry movements while branching into the sciences, the invention of photography, and beyond.
Hochschild's account of all this is certainly testimony to the smallness of Britain's intelligentsia then, but also to the largeness of ideas about human freedom that were moving through and then beyond these networks, the ideas and passions that constitute the atmosphere of an age. Of all the networks he deals with, this one made up of ideas and ethical stirrings is the most important and the most nebulous. The changed spirit and beliefs that link these people in the first place are explicable up to a point and then ultimately mysterious. Why is it that suddenly slavery, which had existed in one form or another throughout history, becomes urgently intolerable not only to the slaves but to privileged people an ocean away from most of the suffering in Africa and the Americas? What had made the Zong's first mate testify against his captain about the murder of those slaves? What made a Cambridge student abandon his career in the church and give his life over to a cause? What made tens or hundreds of thousands of anonymous Britons give up sugar, take up letter-writing and committee-meeting? The networks can be traced, but the stirrings remain mysterious.
Without popular opinion at least periodically rising to meet them, Clarkson and the Quakers would have just been eccentrics and historical footnotes, the rebellious slaves a sad side-story, rather than begetters of a new era. Bury the Chains quotes Wilberforce as writing in his diary "How popular Abolition is, just now! God can turn the hearts of men." But it's clear that it was other men and women, uprisings and revolts, books and pamphlets that did the turning, that the change was mysterious, magnetic or catalytic, but far from divine.
In both Britain and the United States, women who became involved in the antislavery movement began to question the enslavement of their gender, and so goes another long trajectory of links and steps in the expansive history of human rights these last two centuries. I have been reading another book lately, still in manuscript, my friend Susan Schwarzenberg's Becoming Citizens: Family Life and the Politics of Disability. The book traces a group of Seattle-era mothers from the birth of their mentally disabled children to the discovery that their children were denied access to public education to those mothers' engenderment of an educational rights movement. That movement, with interim victories in Washington State, culminated in the 1975 IDEA — Individuals with Disabilities Education Act — some decades after these women decided to change the world to make room in it for children like theirs, not quite two centuries after those nine Quakers and Clarkson met to launch a human-rights movement. The two stories are akin, with initial private moments of realization, the interim building of public associations and communities, and eventually the overhauling of society. Bury the Chains is a kind of template for how the world gets changed, sometimes, for the better. Hochschild's book, like his King Leopold's Ghost before it, often reads like an exciting novel, as one character chases another, or an idea, or a ballot issue across the years. But it pauses periodically to take in the larger landscape of change, and its original subtitle was The First International Human Rights Movement.
The book begins with a kind of trumpet cry: "To understand how momentous was this beginning, we must picture a world in which the vast majority of people are prisoners. Most of them have known no other way of life. They are not free to live or go where they want…. They die young. They are not chained or bound most of the time, but they are in bondage, part of a global economy based on forced labor. Such a world would, of course, be unthinkable today. But that was the world — our world — just two centuries ago, and to most people then, it was unthinkable that it could ever be otherwise. At the end of the eighteenth century, well over three quarters of all people alive were in bondage of one kind or another, not the captivity of striped prison uniforms, but of various systems of slavery or serfdom."
Midway through his story, he halts the narrative to cast about for a reason why the British should have been so much more ready than, say, the French to oppose literal slavery. Here, Hochschild lands upon the essential enslavement of sailors in both the British Navy and its merchant marine, with their press gangs, beatings, kidnappings, horrific conditions and high mortality, which kept the empire whole and the slave trade going. But why should empathy have been extended from sailors to slaves? The answer would make this into another, more speculative book.
The book that Hochschild gives us is valuable instead for its magnificent portrait of how activism works — by coincidences, friendships, patience, and stubbornness, by carefully built networks and belief systems that change slowly or suddenly like climate or the weather. There is the protracted timeline of change: a preliminary state in which almost no one cares about slaves; another moment when it seems like everyone in England does; moments during the Napoleonic wars when it seems like everyone except a few diehards is too frightened — by their own government more than threats from abroad — to say anything about slavery at all; then decades more to go until a final victory. There are interim victories. There are moments of despair. Most of all there are people giving over their lives to a battle that turns out to take more than a lifetime for most of them. And then there are the arguments over how the history will be written — Wilberforce's sons tried to write Clarkson out of it, and succeeded until 1989, when biographer Ellen Gibson Wilson revived his stature as the pivotal figure in the antislavery movement.
You can think of the nuclear freeze movement, which in 1982 had a million proponents gathered on its behalf in New York's Central Park, though few of those stuck with it long enough to realize the "peace dividend" that the collapse of the Soviet Empire was supposed to spawn, or to push further the opportunities for disarmament that arose. The current bout of nuclear proliferation can be blamed in part on Bush, but it is due as well to those who expected a three-year struggle rather than a sixty-year one; any eventual victories will be due in large part to the dedicated minority who have not been realistic, not gone home, not succeeded yet — but might. Or think of the anti-apartheid movement, which like the anti-slavery movement two centuries before combined the nonviolent and the violent, governmental and citizen action, domestic and foreign action, boycotts and educational campaigns to dismantle, piece by piece, slowly, with setbacks, a racist regime (but which, with the moderation that made victory possible, though far less of a victory, never dismantled the extreme financial injustices some call economic apartheid). That story, however, is still unfinished. So, for the record, is the global history of slavery. And what was once the British and Foreign Antislavery Society, founded in 1839 to continue the good work after the signal victory of the year before, is still active as Antislavery International, based in Thomas Clarkson House in London.
Rebecca Solnit lives in a dense atmosphere of conspiracies, digressions and marginalia somewhere in San Francisco. Her most recent book, speaking of conspiracy, is Hope in the Dark: Untold Histories, Wild Possibilities.
Copyright C2004 Rebecca Solnit
This piece first appeared at Tomdispatch.com, with an introduction by Tom Engelhardt.


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The last of the Quilt Market 2008 photos!
This picture originally had Mary and Elizabeth.. however Elizabeth has begged to be 'shopped out. This is the best I can do!
I think we were talking about our phones, I showed off my Sidekick and Mary showed off the coffin phone. She needs the scary ringtone that Camille has to go with the phone! SPOOKY.
This is Denise's booth. I kept going by before market to connect up with her, and would miss her. I don't know how I managed to not get a photo of her.. but here's her lovely booth. I've seen her work up close, she's a local girl!
This is not the best shot ever of the Lakehouse booth. I know Holly will have more photos later.
Here's Camille and Bonnie, her mother. They have a new line of fabric named Cotton Blossoms, which is completely lovely. What fun it must have been for this mother-daughter team.
I can't believe I only got one clear not blurred photo of Joanna's booth. Can that girl rock a spool of string? Check her blog for good photos!
Here's a blog reader, Kathy from The Teacher's Pet. She's not only a pattern designer..
she has her own line of fabric from Troy. Seriously Kathy, you're the famous one!
Check out this adorable little pincushion she gave me! It's in a spot of honor in my sewing room.
Kathy introduced me to her friend Debbie. I think everyone will recognize her.. she's the Granola Girl. It was nice of Kathy to introduce me to her Troy pals!
So that's the end of the people posts friends! I leave you with this view of the Made in Oregon sign we'd pass each day at the end of the Burnside Bridge. Monica, get busy on the little stitchery involving this design, ok?
More of the trip.. coming soon!


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Even as Enrique Peña Nieto basks in a largely successful first year as president, capped off with a massive energy reform that will introduce elements of privatization and foreign investment to Petróleos Mexicanos (Pemex), the state oil company, and the first step of tax reform that will raise VAT of junk food and sodas, Mexicans aren't sure that his administration is making the same progress on security.
Nowhere is that more true than in Michoacán.
A sprawling Pacific state that unfurls from the western Mexican coast inland nearly to the capital of México City, Michoacán wasn't necessarily predestined to become a synonym of drug-fueled anarchy. It's not home to the Zapatista-style insurgency that former president Ernesto Zedillo faced in Chiapas in the mid-1990s, the destabilizing political protests that former president Vicente Fox faced in Oaxaca in 2006, or to the horrific body counts in Ciudad Juárez and elsewhere in Chihuahua that dominated gory headlines just a few years ago during the presidency of Felipe Calderón.


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As the hemp plant keeps gaining popularity, more developments regarding its potential health benefits emerge. Delta-8 is among the latest development that is taking over the market. This compound is associated with intoxicating effects, and most people wish to avoid it. Herein are the tips on stopping a delta 8 high, including taking low concentrations, taking a lot of water, and avoiding cross-contamination.
Delta-8 is made from tetrahydrocannabinol (THC), one of the major chemical compounds in hemp, cannabis, and marijuana. These plants also have other organic chemical compounds such as flavonoids, terpenes, and cannabidiol (CBD). However, unlike other elements, THC receives a lot of attention due to its potential risks. Some states are yet to legalize cannabis products, even for therapeutic benefits because of THC in hemp-based products. If you have been getting "high" with Delta-8 THC, what do you do to stop it?
Are Delta-8 Products Legal In the USA?
According to the 2018 Farm Bill, all hemp-based products are legal within the USA. However, the bill has regulations on producing these products for their user's safety. Although the bill legalized hemp and marijuana-based products in the US, mandates were given to state governments to determine their usage and legality within their territories. As a result, 12 states are yet to legalize hemp-based products, including Delta-8 THC, due to their intoxicating effects.
The 2018 Farm Bill did not legalize marijuana, a cannabis-based plant, because of its high THC levels. Hemp and cannabis sativa were legalized because their THC levels are below 0.3 %. Based on this law, Delta-8 THC products are only legal within the USA if their THC levels are below 0.3%. It is important to
Can Delta-8 Make You Fail A Drug Test?
According to Kale (2019), Delta-8 THC products are likely to make you fail a drug test since it comprises THC as the main element. However, there are various ways of evading this predicament. Always consider the concentrations of Delta-8 THC. Also, avoid marijuana-based Delta-8 products since the THC levels in marijuana are high and will make you fail the test. Lastly, take a lot of water before the drug test. Although this might not be the most effective way, water speeds digestion, clearing out THC from your system faster.
Tips On Avoiding the THC' High'.
Although THC is one of the major chemical compounds of Delta-8 products, there are various ways you can avoid getting high from it, including;
Taking Low Concentrations
THC potency levels should be the priority while buying Delta-8 products. Fattore, Liana & Walter (2011) suggested that concentrated THC has psychoactive side effects that can cause dizziness, poor memory, dry lips, and red eyes. Beginners should be careful even with low concentrations since their body is yet to adapt to THC. They are recommended to begin with low potency levels of 5 mg per serving as they monitor the possible outcome. Veterans and novices shouldn't over-rely on the posted potency levels. The Foods and Drugs Administration (FDA) has conflicted with multiple hemp-based brands for posting wrong potency levels, which expose their customers to overdosing risks.
Using Hemp-based Delta-8 THC
As noted, THC can be obtained from organic hemp, cannabis sativa, or marijuana. However, the THC levels in marijuana are different from those in cannabis sativa and hemp plants. Marijuana has over 2.4% THC levels, which is intoxicating. Therefore, you are likely to get 'high', even with minimum quantities of marijuana-based Delta-8 than hemp or cannabis-based. It is difficult to determine whether delta 8 product is hemp or marijuana-based, and that's when the issue of brand transparency steps in. If the brand you wish to buy your Delta-8 product from has been receiving warnings from the FDA regarding their transparency on potency levels, it's better to avoid it. Such brands can misinform you about their products' ingredients, making you abuse marijuana.
Take A Lot of Water
Although this might not help much, it might make the situation better. Don't panic if you have taken too much Delta-8 THC and wish to minimize the intoxicating side effects, don't panic. Water clears THC from your system faster since it speeds metabolism. According to Morris Robert et al. (2014), alcohol is not an alternative for dehydration in such cases. Instead of it improving the situation, it worsens.
Avoid Cross-Contamination
Cross-contamination is one of the popular ways Delta-8 users experience the "high" effect. Using hemp-based Delta-8 alongside other intoxicating products such as smoking marijuana or sniffing cocaine heightens the chances of getting "high". If you must use medication, check on its possible side effects. If intoxication is among the possible outcomes, avoid it since it intensifies the "high" effect of Delta-8 THC, even from the hemp plant.
Minimize Secondary Exposure to Other Drugs
Although you might have been taking Delta-8 THC responsibly, sharing a room with people smoking marijuana or other drugs increases the chances of getting "high". Kuga et al. (2021) noted that as a passive smoker, you would take a considerable amount of THC, increasing the psychoactive side effects of Delta-8 THC. Avoiding secondary exposure to other drugs keeps you safe, especially if you use Delta-8 THC for therapeutic use, such as lowering anxiety, as suggested by Fattore, Liana & Walter (2011).
Getting high from Delta-8 THC after taking the necessary precautions might be the worst feeling. To avoid such cases, take personal precautions by affirming the actual potency levels of Delta-8, avoiding cross-contamination and secondary exposure to THC, and taking a lot of water to speed digestion. In addition, always check whether the brand you are getting your Delta-8 products from has a clean reputation with the FDA regarding the safety of their products in terms of potency and purity levels. Some brands expose their customers to overdosing unknowingly. Even then, there can be a variance in potency levels, but it shouldn't exceed 10%.
Fattore, L., & Fratta, W. (2011). Beyond THC: The New Generation Of Cannabinoid Designer Drugs. Frontiers In Behavioral Neuroscience, 5, 60.
Kale, N. (2019). Urine Drug Tests: Ordering And Interpreting Results. American Family Physician, 99(1), 33-39.
Kuga, K., Ito, K., Chen, W., Wang, P., Fowles, J., & Kumagai, K. (2021). Secondary Indoor Air Pollution And Passive Smoking Associated With Cannabis Smoking Using Electric Cigarette Device–Demonstrative In Silico Study. Plos Computational Biology, 17(5), E1009004.
Morris, R. G., Teneyck, M., Barnes, J. C., & Kovandzic, T. V. (2014). The Effect Of Medical Marijuana Laws On Crime: Evidence From State Panel Data, 1990-2006. Plos One, 9(3), E92816.]
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