Canonical Adiabatic Theory

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.