Generator, canonical transformation

Readers familiar with canonical transformation theory [37] can confirm that these results follow from use of a type 4 generating function,... 

An alternative procedure to generate a canonical transformation is to use the Hamiltonian flow itself. Consider an arbitrary Hamiltonian system of the same dimension as the original system. The associated functional dependence of the final state ait = tf on the initial state at t = t, can be represented by... 

We see now that the indeterminacy of the Lagrangian is very important for the existence of canonical transformations. The function F in (3.1.31) is called the generating function of the canonical transformation. 

This means that the canonical transformation is determined once Fi, the generating function, is known. 

Of particular importance to the forthcoming development is the statement of the change induced in a property A(p, q) by the generator of an infinitesimal canonical transformation G as expressed in the Poisson bracket notation. The change in the property A obtained as a result of an infinitesimal canonical transformation of the coordinates is given by... 

Earlier in this section it was commented on how the minimal-coupling QED Hamiltonian is obtained from fhe classical Lagrangian function. A few words are in order regarding the derivation of the multipolar Hamiltonian (6). One method involves the application of a canonical transformation to the minimal-coupling Hamiltonian [32]. In classical mechanics, such a transformation renders the Poisson bracket and Hamilton s canonical equations of motion invariant. In quantum mechanics, a canonical transformation preserves both the commutator and Heisenberg s operator equation of motion. The appropriate generating function that converts H uit is propor-... 

Let (p, q) be one set of canonically conjugate coordinates and momenta (the old variables) and (P,Q) be another such set (the new variables).13 (P, Q, p and q are IV-dimensional vectors for a system with N degrees of freedom, but for the sake of clarity multidimensional notation will not be used the explicitly multidimensional expressions are in most cases obvious.) In classical mechanics P and Q may be considered as functions of p and q, or inversely, P and Q may be chosen as the independent variables with p and q being functions of them. To carry out the canonical transformation between these two sets of variables, however, one must rather choose one old variable and one new variable as the independent variables, the remaining two variables then being considered as functions of them. The canonical transformation is then carried out with the aid of a generating function, or generator, which is some function of the two independent variables, and two equations which express the dependent variables in terms of the independent variables.13... 

There are, just as in classical mechanics, four ways of choosing one old variable and one new variable , so there are four equivalent sets of unitary transformation elements connecting the old and new representations << Q>, (q Py,

and

. The fundamental correspondence relations express the classical limit of these unitary transformation elements in terms of the classical generating functions for the related classical canonical transformation ... 

The classical generator which effects this canonical transformation is ... 

Let us go back to the general problem of dynamics. The goal is to perform a near the identity canonical transformation that gives the Hamiltonian a suitable form, that we shall generically call a normal form. We shall use the method based on composition of Lie series. Let us briefly recall how the method works. A near the identity transformation is produced by the canonical flow at time e of a generating function %(p, q), and takes the form... 

Thus, in order to give the Hamiltonian the normal form (7) we should remove the unwanted terms A(q), B(q),p). Remark that these terms are small provided / is small. Following Kolmogorov, we look for a canonical transformation with generating function... 

We perform a first canonical transformation with generating function X i q) = X l q) + ( W, q), where the function X(q) and the real vector are determined from the equations... 

Let us consider a canonical transformation with generating function of the form... 

S may be regarded as the generator of a canonical transformation instead of S. The equation... 

It is only the multiply-pcriodic solutions which are of importance for the quantum theory. The methods which we shall employ for their deduction in what follows are essentially the same as those which Poincard has treated in detail in his Methodes nouvdles de la Mdtanique celeste,2 By a solution we mean, as usual, the discovery of a principal function S which generates a canonical transformation,... 

To find them, we have to look for the generator S(w°, J) of a canonical transformation... 

The angle and action variables of the unperturbed motion, in this case that of the harmonic oscillator, are given by the canonical transformation with the generator (cf. (16), 7)... 

This may be accomplished by means of a canonical transformation with the generator... 

Another, slightly more technical ( ) way to understand area preservation is to recall that the coordinates of all the points on the Poincare map are specified by coordinates that are canonically conjugate. Because both the initial and final coordinates of a family of trajectories propagated for one mapping are so specified, there must exist a generating function that transforms the coordinates of the initial points into those of the final points. Such a generating function is necessarily a canonical transformation. All canonical transformations preserve the norm of the vectors they transform it can be shown that this property is equivalent to area preservation on the Poincare map. ... 

First we consider a system with two degrees of freedom (N = 2). Suppose we have two closed curves yi and y2 phase space, both of which encircle a tube of trajectories generated by Hamilton s equations of motion. These curves can be at two sequential times (tj, or they can be at two sequential mappings on a Poincare map These curves are associated with domains labeled (Dj, D2), which are the projections of the closed curves upon the coordinate planes (pj, qj. Because both the mapping and the time propagation are canonical transformations, the integral invariants (J-j,. 2) are preserved (constant) in either case. There are two of them, of the form ... 

The generating function T(, , ) for the nonautonomous canonical transformation may be computed from the following indefinite integral ... 

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