Titlebook: Classical and Quantum Dynamics; From Classical Paths Walter Dittrich,Martin Reuter Textbook 20013rd edition Springer-Verlag Berlin Heidelbe

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The Adiabatic Invariance of the Action Variables,We shall first use an example to explain the concept of adiabatic invariance. Let us consider a “super ball” of mass ., which bounces back and forth between two walls (distance .) with velocity ... Let gravitation be neglected, and the collisions with the walls be elastic. If .. denotes the average force onto each wall, then we have

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Superconvergent Perturbation Theory, KAM Theorem (Introduction),Here we are dealing with an especially fast converging perturbation series, which is of particular importance for the proof of the KAM theorem (cf. below).

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Examples for Calculating Path Integrals,We now want to compute the kernel .) for a few simple Lagrangians. We have already found for the one-dimensional case that . with

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Yichao Lu,Ruihai Dong,Barry Smythparticular, we want to investigate the conditions under which a path is a minimum of the action and those under which it is merely an extremum. For illustrative purposes we consider a particle in two-dimensional real space. If we parametrize the path between points . and . by ?, then Jacobi’s principle states:

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