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

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https://doi.org/10.1007/978-3-642-85278-7rs appear in the expression for the adiabatic invariants. We now wish to begin to locally remove such resonances by trying, with the help of a canonical transformation, to go to a coordinate system which rotates with the resonant frequency.

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Jacobi Fields, Conjugate Points,particular, 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 ϑ, then Jacobi’s princi

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Action-Angle Variables,, ..) is the generator of a canonical transformation to new constant momenta .., (all .., are ignorable), and the new Hamiltonian depends only on the ..,: . = . = .(..). Besides, the following canonical equations are valid:

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Time-Independent Canonical Perturbation Theory, conservative, ∂./∂. = 0, and periodic in both the unperturbed and perturbed case. In addition to periodicity, we shall require the Hamilton-Jacobi equation to be separable for the unperturbed situation. The unperturbed problem ..(..) which is described by the action-angle variables .. and .. will b

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Removal of Resonances,rs appear in the expression for the adiabatic invariants. We now wish to begin to locally remove such resonances by trying, with the help of a canonical transformation, to go to a coordinate system which rotates with the resonant frequency.

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The KAM Theorem,rator .(.,., ..) converges (according to Newton’s procedure) and thus the invariant tori are not destroyed. The KAM theorem is valid for systems with two and more degrees of freedom. However, in the following, we shall deal exclusively with the case of two degrees of freedom.