Lie transforms

S. Martfnez-Garaot, E. Torrontegui, X. Chen, and J. G. Muga Shortcuts to adiabaticity in three-level systems using Lie transforms. Phys. Rev. A, 89(5) 053408—053415(2014). 

Briefly, the aim of Lie transformations in Hamiltonian theory is to generate a symplectic (that is, canonical) change of variables depending on a small parameter as the general solution of a Hamiltonian system of differential equations. The method was first proposed by Deprit [75] (we follow the presentation in Ref. 76) and can be stated as follows. 

Now, we apply Lie perturbation theory to the Hamiltonian (15) to derive the Fenichel normal form. In the Lie transformation, we use the variables (z, z) for convenience. For the vibrational degrees of freedom, (zn,Zn) are the following complex conjugate variables ... 

We will use the Lie transformation where F in Eq. (20) is given by a polynomial yi, which consists of feth-order terms as follows ... 

Using Fi for k which is greater than or equal to 3, we define a series of Lie transformations as follows ... 

Therefore, the Lie transformation of the second-order terms is given by... 

X = pj can be eliminated by the Lie transformation—that is, by suitably choosing the coefficients f in the second equation of Eq. (24). This means that the degree of freedom z, zi), which corresponds to the reaction coordinate, only appears as a product of z Zi in the transformed Hamiltonian K(z,z) as follows ... 

Thus, all the terms that cannot be eliminated by the Lie transformation are products of the form z2p, , kvP- By noting the relation... 

Thus, the Lie transformation brings the Hamiltonian locally near a saddle with index 1 into the Fenichel normal form. In addition, we find that, on the NHIM, tori with sufficiently nonresonant frequencies survive. 

For saddles with index L with L > 1, note the following point when we use the Lie transformation. For these saddles, one may wonder whether resonance can occur among the normal directions in the following sense. If a set of nonnegative integers / (/ = I,..., L) exists where all of its elements are not zero, such that Yl i= holds, then resonance occurs among... 

For saddles where resonance does not take place, we can use the Lie transformation in a similar way to saddles with index 1. Then, only those terms... 

Let X = q,p) denote the one-degree-of-freedom reaction coordinate. For M-degrees-of-freedom vibrational modes, 7 e R" and 0 G T" denote their action and angle variables, respectively, where T = [0,27t]. These action and angle variables would be obtained by the Lie transformation, as we have discussed in Section IV. In reaction dynamics, the variables (/, 0) describe the degrees of freedom of the intramolecular and possibly the intermolecular vibrational modes that couple with the reaction coordinate. In the conventional reaction rate theory, these vibrational modes are supposed to play the role of a heat bath for the reaction coordinate x. 

Since the manifold Mq is a NHIM, it changes continuously, under a small perturbation, into a new NHIM M - Moreover, the separatrix Wq changes, continuously and locally near M, into the stable manifold and the unstable one W" of the NHIM M. Note, however, that, in general, and W no longer coincide with each other to form a single manifold globally. Then, the Lie transformation method brings the total Hamiltonian H x.I, 0) into the Fenichel normal form locally near the manifold M. ... 

---