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Think of one particle of a layer. It is moving in simple harmonic motion around the center, since $$a\propto x$$ Now think of the whole layer. The whole layer is moving together. If you can analyze how one particle of this layer moves than you know how the whole layer moves. If you understand how each layer moves then you understand how the whole sphere ...

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Well, given $$\dot{p} = - \frac{\partial H}{\partial x} \quad \text{and} \quad \dot{q} = \frac{\partial H}{\partial p}$$ we have $$H = - \int dx_{i} \, g_{i}(x,p,t)$$ and $$H = \int dp_i \, f_{i} (x, p, t).$$ You deal with the constants of integration using the obvious constraint: $$- \int dx_{i} \, g_{i} (x, p, t) = \int d p_{i} \, f_{i} (x,... 2 Assuming from the notation$$ \dot{x}^i~=~f^i(x,p,t), \qquad \dot{p}_i~=~g_j(x,p,t), \tag{1}$$that the symplectic structure is the standard canonical symplectic structure$$\omega = \sum_{i=1}^n\mathrm{d}p_i\wedge \mathrm{d}x^i,\tag{2}$$we get that$$\begin{align}\mathrm{d}H(x,p,t)- \frac{\partial H(x,p,t)}{\partial t}\mathrm{d}t ~=~&\sum_{i=1}^n\...

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By checking the relevant reference (and iterating once) you arrive at the paper by Qi Ouyang, Harry L. Swinney, and Ge Li, "Transition from Spirals to Defect-Mediated Turbulence Driven by a Doppler Instability" (Phys. Rev. Lett. 84, 1047; e-print 1, 2), which describes: The observed spiral instability occurs whent he spiral tip meanders and the Doppler ...

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I don't know if I can state a clear single "definition", but hopefully I will be able to sort out some of the concepts and the confusion. integrability is sometimes associated with having a closed form solution This, I think, is categorically not true. At least in the usual sense of 'closed form'. If you take the Lieb-Liniger model, which is I believe ...

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After doing some studies on this subject for months now I did received the answer to the question that I've been looking out for(I'm really Shameful for myself now).Ok then, here it goes as the phenomena of Chaos is Unpredictable so one can NEVER determine the exact values of Chaotic System Parameters for which the system exhibit Sensitive dependence on ...

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