Download Geography of Order and Chaos in Mechanics: Investigations of by Bruno Cordani PDF

By Bruno Cordani

This unique monograph goals to discover the dynamics within the specific yet vitally important and demanding case of quasi-integrable Hamiltonian structures, or integrable platforms a bit perturbed by means of different forces. With either analytic and numerical tools, the e-book stories a number of of those systems—including for instance the hydrogen atom or the sunlight procedure, with the linked Arnold web—through smooth instruments comparable to the frequency changed fourier remodel, wavelets, and the frequency modulation indicator. in the meantime, it attracts seriously at the extra usual KAM and Nekhoroshev theorems.

Geography of Order and Chaos in Mechanics should be a necessary source for pro researchers and likely complex undergraduate scholars in arithmetic and physics, yet regularly may be an outstanding reference for Ph.D. scholars with an curiosity in perturbation theory.

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Additional resources for Geography of Order and Chaos in Mechanics: Investigations of Quasi-Integrable Systems with Analytical, Numerical, and Graphical Tools

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40 Analytical Mechanics and Integrable Systems We can rephrase the content of this section in terms of left (right) invariant dual forms. Consider the vector space g∗ , the dual space of g, and let {ωa } be the dual basis of {Lb }, that is, ωa , Lb = δa b ; then one can show that a dωa + cbc ωb ωc = 0, a, b, c = 1, . . , N. These equations are called the Maurer–Cartan equations for G. The ωa are left invariant forms, as one easily deduces by applying the general relation α, φ∗ v = φ∗ α, v : a ωa , Lb = ωa , λx∗ Lb = λ∗ x ω , Lb a a ⇒ λ∗ xω = ω .

Jsr = (A−1 )h11 . . (A−1 )hrr Aj11 . . h for the corresponding change of the components. , summing over) a pair of such indices leads to a quantity invariant under change of basis. ” A tensor field is the assignment to every point p ∈ M of a tensor whose underlying vector space is Tp M; thus eh = ∂h and ek = dx k . The definition of push-forward and pull-back is naturally extended to the tensor fields, h with Ah k = (Tφ )k . Forms and Exterior Derivative Let us define the symmetry properties of a tensor.

Assume for simplicity n = 3. 2 on page 116 a section with the plane ω3 = 1 in the 3-dimensional frequency space is shown, thus with equation k1 ω1 + k2 ω2 + k3 = 0 : the “skeleton” is formed by the lines whose slope and intersection with the axes take rational values, “fleshed out” by the resonance strips. The Arnold web is connected, open, and dense in the action space with, however, a relative small measure vanishing with the square root of the perturbative parameter. 2 appears, distorted under the diffeomorphism given by the local inverse of the frequency map I → ω(I).

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