By Sandro Wimberger

ISBN-10: 3319063421

ISBN-13: 9783319063423

ISBN-10: 331906343X

ISBN-13: 9783319063430

The box of nonlinear dynamics and chaos has grown greatly during the last few many years and is changing into increasingly more suitable in numerous disciplines. This publication offers a transparent and concise creation to the sphere of nonlinear dynamics and chaos, appropriate for graduate scholars in arithmetic, physics, chemistry, engineering, and in traditional sciences quite often. It presents a radical and sleek creation to the strategies of Hamiltonian dynamical structures' thought combining in a accomplished approach classical and quantum mechanical description. It covers a variety of issues often now not present in related books. Motivations of the respective matters and a transparent presentation eases the knowledge. The ebook is predicated on lectures on classical and quantum chaos held by means of the writer at Heidelberg collage. It comprises routines and labored examples, which makes it perfect for an introductory direction for college students in addition to for researchers beginning to paintings within the field.

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**Additional info for Nonlinear Dynamics and Quantum Chaos: An Introduction**

**Sample text**

Sufficient regularity: A smoothness condition for the perturbation (sufficient number of continuous derivatives of H1 ) has to hold. This condition guarantees that the Fourier coefficients of expansions as in Eq. 23) decrease sufficiently fast. For more details consult [22, 24]. 34) where γ is a constant that depends only on ε, on the magnitude of the perturbation H1 and on some measure of the nonlinearity of the unperturbed Hamiltonian H0 . The exponent χ depends on the number of degrees of freedom and the regularity properties of H1 .

Here are some simple examples: 1. Mass point in an external potential: 1 A separatrix is a boundary in phase space separating two regions of qualitatively different motions, see Fig. 1 below for the pendulum. 2 To simplify the notation, vector quantities are not always specified in boldface. From the local context it should be clear that the corresponding symbols represent n-dimensional vectors. 26 3 Nonlinear Hamiltonian Systems H= p2 + V (q). 2) The equations of motion are equivalent to Newton’s second law m q¨ = −ς V /ςq and read ςH p ςH ςV q˙ = = , p˙ = − =− .

20) We note that the determinants in the numerator and in the denominator of Eq. 19) equal each other because they can be brought into the same form by interchanging rows and columns. After having shown that canonical transformations preserve the phase space volume we demonstrate that the temporal evolution itself is generated by canonical transformations. Let us assume we are given the solution of the Hamiltonian equations of motion. We define the old and the new coordinates by q0 = q, p0 = p and Q(q, t) = q(t), P( p, t) = p(t).

### Nonlinear Dynamics and Quantum Chaos: An Introduction by Sandro Wimberger

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