Transformation to Canonical Form

Numerical integration of ordinary differential equations is most conveniently performed when the system consists of a set of n simultaneous first-order ordinary differential equations of the form  

This is called the canonical form of the equations. When the initial conditions are given at a 

The above problem can be condensed into matrix notation, where the system equations are represented by 

Differential equations of higher order, or systems containing equations of mixed order, can be transformed to the canonical form by a series of substitutions. For example, consider the th-order differential equation 

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This cubic can he transformed to canonical form (with the quadratic term missing) hy means of the linear transformation... 

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. 

We can find the canonical forms ourselves. To evaluate the observable canonical form Aob, we define a new transformation matrix based on the controllability matrix ... 

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... 

This solution reduces/from 28 to —8. The immediate objective is to see if it is optimal. This can be done if the system can be placed into feasible canonical form with x5, 3, —/ as basic variables. That is, 3 must replace xx as a basic variable. One reason that the simplex method is efficient is that this replacement can be accomplished by doing one pivot transformation. 

Assuming that a polynomial has been found which adequately represents the response behavior, it is now possible to reduce the polynomial to its canonical form. This simply involves a transformation of coordinates so as to express the response in a form more readily interpreted. If a unique optimum (analogous to a mountain peak in three dimensions) is present, it will automatically be located. If (as is usual in multidimensional problems) a more complex form results, the canonical equation will permit proper interpretation of it. 

Having obtained an adequate mathematical second-order model of a research subject for k<3, we can obtain a geometric interpretation in a two- or three-dimensional space. To obtain this interpretation, it is necessary to transform the second-order model into a typical-canonical form. Canonic transformation of a regression model is terminated in the form ... 

For k<3, it is possible, after canonic transformation, to determine to which type of geometric surface the given equation corresponds. It is known from mathematical analysis that there exist 17 second-order surfaces of standard form. By a canonic regression model, we can make this sorting by extreme types ... 

According to values of regression coefficients it is evident that factor X2 has the smallest effect on response. The existence of triple interaction makes transformation of the regression model into canonical form difficult, which, in turn, makes a detailed study of the response surface and establishing conditions for maximal response impossible. 

The product of a C x Ns matrix and a JVS x 1 matrix is a C x 1 matrix note that Ns disappears as one of the dimensions of the resultant matrix. The amounts of components in a reaction system are independent variables and consequently do not change during a chemical reaction. The amounts of species are dependent variables because their amounts do change during chemical reactions. Equation 5.1-27 shows that A is the transformation matrix that transforms amounts of species to amounts of components. The order of the columns in the A matrix is arbitrary, except that it is convenient to include all of the elements in the species on the left so that the canonical form can be obtained by row reduction. When the row-reduced form of A is used, the amounts of the components CO, H2, and CH4 can be calculated (see Problem 5.1). 

In this / -matrix theory, open and closed channels are not distinguished, but the eventual transformation to a A -matrix requires setting the coefficients of exponentially increasing closed-channel functions to zero. Since the channel functions satisfy the unit matrix Wronskian condition, a generalized Kohn variational principle is established [195], as in the complex Kohn theory. In this case the canonical form of the multichannel coefficient matrices is... 

We concentrate on the expression for the covalent part of the crystal field Wffff and transform it to the form coinciding with the AOM and relate the parameters of the latter with the electronic structure of the ligands. To do so, we perform a unitary transformation of the canonical MOs of the /-system Z) - the eigenstates of the Fock operator - to the localized MO L) separately for the occupied and vacant canonical MO ... 

As seen in Equation 8.10, there is a linear dependence between the input variables or controlled factors that create a nonunique solution for the regression coefficients if calculated by the usual polynomials. To avoid this problem, Scheffe [3] introduced the canonical form of the polynomials. By simple transformation of the terms of the standard polynomial, one obtains the respective canonical forms. The most commonly used mixture polynomials are as follows ... 

In summary, the model allows for two types of interactions between the mirror spaces, the weak kinematical perturbation and the adiabatic and sudden limits equivalent to Eq. (17) or Eqs. (29)-(34). The overwhelming rate of particles over antiparticles in the Universe is inferred in this picture once the particular particle state has been selected. The Minkowski metric of the special theory of relativity is represented here by a non-positive definite metric, Eq. (8), bringing about a quantum model with a complex symmetric ansatz. Although the latter permits general symmetry violations, it is nevertheless surprising that fundamental transformations between complex symmetric representations and canonical forms come out unitary. 

It is illustrative to study these properties also by using the matrix representations, in which case one can also generalize the results to degenerate eigenvalues. Starting from (1.17), one knows that the matrix T may be brought to classical canonical form X by a similarity transformation y, so that... 

Let us start from a linearly independent set = Ot, 2,..., complex functions the set has the additional property that the overlap matrix A = <0 fl>) is nonsingular, i.e., that A 0. Let y be the similarity transformation, which brings A to classical canonical form k with the eigenvalues on the diagonal and Os and Is on the line above the diagonal ... 

If the transformation pathway cannot be reduced to monomolecular reactions, nonunit stoichiometric coefficients may appear at some junction points of the kinetic resistors. In terms of electric circuits, this means that the absence of the balance of the current inflow and outflow at this June tion point may cause norJinearity and deviations from the canonical form of the KirchhofF equation. 

Finding canonical forms for each of the A s, is an easier task than for any general form A. A technique has been developed at Princeton [189-191] for finding such canonical forms using an algebraic method in nonlinear perturbation theory. For the application of this technique the leading operator Aq must be chosen to be in canonical form. The basic approach is then to find a non-linear transformation C such that the resultant operator. 

The first stage in the process is to convert Ao to a suitable canonical form. This is achieved by using the transformation from the original fast variables to some new, purely fast, variables [Pg.399]

The state of a classical system is specified in terms of the values of a set of coordinates q and conjugate momenta p at some time t, the coordinates and momenta satisfying Hamilton s equations of motion. It is possible to perform a coordinate transformation to a new set of ps and qs which again satisfy Hamilton s equation of motion with respect to a Hamiltonian expressed in the new coordinates. Such a coordinate transformation is called a canonical transformation and, while changing the functional form of the Hamiltonian and of the expressions for other properties, it leaves all of the numerical values of the properties unchanged. Thus, a canonical transformation offers an alternative but equivalent description of a classical system. One may ask whether the same freedom of choosing equivalent descriptions of a system exists in quantum mechanics. The answer is in the affirmative and it is a unitary transformation which is the quantum analogue of the classical canonical transformation. 

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