Titlebook: Classical and Quantum Dynamics; From Classical Paths Walter Dittrich,Martin Reuter Textbook 20164th edition Springer International Publishi

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,The Hamilton–Jacobi Equation,We already know that canonical transformations are useful for solving mechanical problems. We now want to look for a canonical transformation that transforms the 2. coordinates (.., ..) to 2. constant values (.., ..), e.g., to the 2. initial values . at time . = 0. Then the problem would be solved, . = .(.., .., .), . = .(.., .., .).

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Action-Angle Variables,In the following we will assume that the Hamiltonian does not depend explicitly on time; .∕. = 0. Then we know that the characteristic function .(.., ..) is the generator of a canonical transformation to new constant momenta .. (all .. are ignorable), and the new Hamiltonian depends only on the ..: . = . = .(..).

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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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Fundamental Principles of Quantum Mechanics,There are two alternative methods of quantizing a system:

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https://doi.org/10.1007/978-3-030-28003-1 ., ., are points in .-dimensional configuration space. Thus ..(.) describes the motion of the system, and . determines its velocity along the path in configuration space. The endpoints of the trajectory are given by ..(..) = .., and ..(..) = ...

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Maintaining Temporal Warehouse Models, translation . and .(..) = 0. Then the noninvariant part of the action, . is given by . and thus it immediately follows for the variation of . that . or . Here we recognize Newton’s law as nonconservation of the linear momentum: . Now it is straightforward to derive a corresponding law of nonconserv

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Katalin Ternai,Szabina Fodor,Ildikó Szabóparticular, 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 princi

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