Bây giờ chúng ta hãy xem xét một vấn đề với Hamilton được bảo tồn trong một ý nghĩa gần đúng bằng một hệ thống khả tích với mức độ tự do n. Hệ thống này có khả tích được mô tả với một H Hamilton (0), và chúng tôi cho rằng chúng tôi đã mô tả nó trong điều khoản của biến hành động của mình (0) (0) Ii và các biến góc φi. | 194 CHAPTER 7. PERTURBATION THEORY so it need not be uniquely defined. This is what happens for example for the two dimensional harmonic oscillator or for the Kepler problem. Canonical Perturbation Theory We now consider a problem with a conserved Hamiltonian which is in some sense approximated by an integrable system with n degrees of freedom. This integrable system is described with a Hamiltonian H 0 and we assume we have described it in terms of its action variables Iị and angle variables . This system is called the unperturbed system and the Hamiltonian is of course independent of the angle variables H0 f 0 0 H0 . The action-angle variables of the unperturbed system are a canonical set of variables for the phase space which is still the same phase space for the full system. We write the Hamiltonian of the full system as H 0 ỹ0 H 0 f 0 eHi 0 ỹ 0 . We have included the parameter e so that we may regard the terms in Hi as fixed in strength relative to each other and still consider a series expansion in e which gives an overall scale to the smallness of the perturbation. We might imagine that if the perturbation is small there are some new action-angle variables ĩ and ội for the full system which differ by order e from the unperturbed coordinates. These are new canonical coordinates and may be generated by a generating function of type 2 . F E c Ii eFl Ợ. ỹm . This is a time-independent canonical transformation so the full Hamiltonian is the same function on phase-space whether the unperturbed or full action-angle variables are used but has a different functional form H H 0 0 . Note that the phase space itself is described periodically by the coordinates Ị 0 so the Hamiltonian perturbation H1 and the generating . CANONICAL PERTURBATION THEORY 195 function F1 are periodic functions with period 2 in these variables. Thus we can expand them in Fourier series H1 ĩ ỉ X H ĩ 0 jW . . . . F1 X F I eikf 0 . k where the sum is over all .
