Transformations canonical

There are several types of transformations that can be applied to matrices. One of the most useful is the canonical transformation. To transform a matrix, you pre-multiply by a matrix of constants and postmultiply by another matrix of constants. The canonical transformation is one which converts a matrix into another matrix that is diagonal and has the eigenvalues of the original matrix as its diagonal elements. 

Consider the system described by the linear, homogeneous ordinary differential equations 

The eigenvalues of the 4 matrix will be the roots of the characteristic equation of the system, and they can be calculated from Eq. (15.18). 

Now suppose we change variables by defining a new vector of variables z according to the following equation. 

Actually, the V matrix consists of the eigen vectors of the 4 matrix. The interested reader is referred to the texts mentioned in Sec. 15.1 for details. For our purposes, we need only to accept the notion that such a matrix of constants can be found that will convert the nondiagonal A into the diagonal 

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The practical way of calculating 2 is different from that used in the derivation of (4.18). Since 2 is invariant with respect to canonical transformations, it is preferable to seek it in the initial coordinate system. Writing the linearized equation for deviations from the instanton solution 6Q,... 

A change of variables, from qi, pi ) to ( Qi, Pi ), is called a canonical transformation, if the transformed equations of motion have the same form as that given in equation 4,36, but with H = H( Qi, Pi )- In particular, if there exists a canonical transformation such that % = H Pi,P2,.., F ) only, then the equations... 

In the Hamiltonian conventionally used for derivations of molecular magnetic properties, the applied fields are represented by electromagnetic vector and scalar potentials [1,20] and if desired, canonical transformations are invoked to change the magnetic gauge origin and/or to introduce electric and magnetic fields explicitly into the Hamiltonian, see e.g. refs. [1,20,21]. Here we take as our point of departure the multipolar Hamiltonian derived in ref. [22] without recourse to vector and scalar potentials. 

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

Once this divergence happens, further solution of the differential equation is not possible beyond this point, and we have to reformulate the problem. To clarify our idea, let us consider the ID problem. At the turning point, p q) = 0 and A diverges. If we invert A to A = dq/dp, the divergence is removed and the propagation of A proceeds smoothly through the caustics. This inversion is equivalent to the canonical transformation, (p,q) (—q,p). It can be easily... 

Let us begin by studying the relative dynamics in the Hamiltonian case, that is, for deterministic driving. The shift (42) to the moving origin is a time-dependent canonical transformation [98]. It transforms the linearized Hamiltonian (33) into... 

A. Deprit, Canonical transformations depending on a small parameter, Cel. Mech. 1,12 (1969). 

That includes transforming a given system to the controllable canonical form. We can say that state space representations are unique up to a similarity transform. As for transfer functions, we can say that they are unique up to scaling in the coefficients in the numerator and denominator. However, the derivation of canonical transforms requires material from Chapter 9 and is not crucial for the discussion here. These details are provided on our Web Support. 

The rest of this section requires material on our Web Support and is better read together with Chapter 9. Using the supplementary notes on canonical transformation, we find that the observable canonical form is the transpose of the controllable canonical form. In the observable canonical form, the coefficients of the characteristic polynomial (in reverse sign) are in the last column. The characteristic polynomial is, in this case,... 

The transformation t we saw at the end of the last section, which changes liberal schemes into free schemes, is such a canonical transformation. The corresponding canonical class of schemes is the class of schemes such that tests are applied initially on the input variables and are applied after assignment statements on the program variables involved, and at no other time. This transformation t is clearly recursive and equivalence preserving. The class of free schemes is not a canonical form class, since, as we saw, there are schemes not strongly equivalent to any free scheme. 

There is no way to change undecidable properties to decidable properties without a loss of power of expression. So that sort of consideration cannot be an argument for or against the use of any particular canonical form. However, when the property happens to be decidable in general or for a particular subclass, use of a particular format may make life easier. Further, although WILE schemes form a canonical form for the whole class of schemes, they do not do so for many subclasses. That is, If we have a canonical transformation... 

COROLLARY 4.21 Any canonical transformation mapping program schemes into WHILE schemes cannot preserve computational equivalence. 

On the other hand, one may perform on the effective Hamiltonian in representation II the canonical transformations ... 

Equation (49) contains the Franck-Condon factors that are the matrix elements of the translation operator involved in the canonical transformation (36) with k = 1 that are given for m > n by... 

We may recall and emphasize that the autocorrelation function obtained in the three representations I, II, and III must be equivalent, from the general properties of canonical transformation which must leave invariant the physical results. Thus, because of this equivalence, the spectral density obtained by Fourier transform of (43) and (45) will lead to the same Franck-Condon progression (51). 

A transformation q,p —> q, p possessing the property that the canonical equations of motion also hold for the new coordinates and momenta, is called a canonical transformation. 

The foregoing unitary transformations may be interpreted as the analogues of canonical transformations in classical mechanics. 

Invariance of the trace of a matrix under unitary transformation corresponds to the invariance of phase density under canonical transformation in classical theory. 

The mathematical details of carrying out such a redefinition of coordinate system, termed a canonical transformation, have been presented elsewhere in detail (D2). 

The canonical transformation is useful in a number of ways. For example, it can be used to solve the system of ordinary difTerential equations... 

Thus the canonical transformation can be used to solve differential equations. It is also useful for other things, as we will see later. 

Let us canonically transform the Q matrix into a diagonal matrix which has as its diagonal elements the eigenvalues of Q. 

It is somewhat similar to canonical transformation. But it is different in that the diagonal 2 matrix contains as its diagonal elements, not the eigenvalues of the Kj, matrix, but its singular values. 

They would become the stars of Prigoginian statistical mechanics. Their importance lies in the fact that, whenever it is possible to determine these variables by a canonical transformation of the initial phase space variables, one obtains a description with the following properties. The action variables / ( = 1,2,..., N, where N is the number of degrees of freedom of the system) are invariants of motion, whereas the angles a increase linearly in time, with frequencies generally action-dependent. The integration of the equations... 

We first consider a Hamiltonian, thus deterministic, system. Denoting by oo the set of all phase space coordinates of a point in phase space (which determines the instantaneous state of the system), the motion of this point is determined by a canonical transformation evolving in time, 7), with Tq = I. The function of time TfCO thus represents the trajectory passing through co at time zero. The evolution of the distribution function is obtained by the action on p of a unitary transformation Ut, related to 7) as follows ... 

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