Titlebook: Classical and Quantum Dynamics; From Classical Paths Walter Dittrich,Martin Reuter Textbook 2020Latest edition The Editor(s) (if applicable

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Kristin L. Huffman,Andrea Giordanors 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 ., then Jacobi’s principle states

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,Poincaré Surface of Sections, Mappings,o-dimensional surface. If we then consider the trajectory in phase space, we are interested primarily in its piercing points through this surface. This piercing can occur repeatedly in the same direction. If the motion of the trajectory is determined by the Hamiltonian equations, then the . + 1-th piercing point depends only on the .th.

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https://doi.org/10.1007/978-3-030-93186-5..) 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