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Canonical Adiabatic Theory

  • Walter Dittrich
  • Martin Reuter

Abstract

In the present chapter we are concerned with systems, the change of which — with the exception of a single degree of freedom — should proceed slowly. (Compare the pertinent remarks about ε as slow parameter in Chap. 7.) Accordingly, the Hamiltonian reads:
$$ H = {H_0}(J,\;\varepsilon {p_i},\;\varepsilon {q_i};\;\varepsilon t) + \varepsilon {H_1}(J,\;\theta {\rm{, }}\varepsilon {p_i},\;\varepsilon {q_i};\;\varepsilon t)\;{\rm{.}} $$
(10.1)
Here, (J, θ) designates the “fast” action-angle variables for the unperturbed, solved problem H0(ε = 0), and the (p i , q i ) represent the remaining “slow” canonical variables, which do not necessarily have to be action-angle variables. Naturally, we again wish to eliminate the fast variable θ in (10.1). In zero-th order, the quantity which is associated to θ is denoted by J. In order to then calculate the effect of the perturbation εH1, we look for a canonical transformation \( (J,\;\theta ,\;{p_i},\;{q_i}) \to (\bar J,\;\bar \theta ,\;{\bar p_i},\;{\bar q_i}) \) which makes the new Hamiltonian \( \tilde H \) independent of the new fast variable θ.

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Copyright information

© Springer-Verlag Berlin Heidelberg 1992

Authors and Affiliations

  • Walter Dittrich
    • 1
  • Martin Reuter
    • 2
  1. 1.Institut für Theoretische PhysikUniversität TübingenTübingenFed. Rep. of Germany
  2. 2.Institut für Theoretische PhysikUniversität HannoverHannover 1Fed. Rep. of Germany

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