The Canonical Equations

Each of Lagrange s equations is a differential equation of the second order. In many cases, particularly for work of a general character, it is desirable to replace them by a system of twice as many differential equations of the first order. The simplest way of accomplishing this is to put qk=sk, and then to take these additional equations into account, treating the sk s, as well as the qks, as unknown quantities. A much more symmetrical form is obtained as follows  

1 A Legendre Transformation transforms, in general, a function f(x, y) into a 

This is the so-called canonical form of the equations of motion. R qx, pv q2, p2. .. t) is called the Hamiltonian function. The variables qt and pk are said o be canonically conjugated. 

The same equations are obtained if the momenta are defined by (1) in the same way, and the function L in the variation principle (2), 4, is expressed in terms of H by means of equation (3). We have 

All of these considerations remain valid if the function L, and with it the function H, depends explicitly on the time t. The latter case will occur, for example, cither if external influences depending on the time are present (U depending on t), or if, in the case of a self-contained Bystem, a system of reference is employed which itself 

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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. 

This differential equation is easily transformed to the canonical equation of Hermite polynomials ... 

As first noted by Dirac [85], the canonical equations of motion for the real variables X and P with respect to J Pmf are completely equivalent to Schrddinger s equation (28) for the complex variables d . Moreover, it is clear that the time evolution of the nuclear DoF [Eq. (32)] can also be written as Hamilton s equations with respect to M mf- Similarly to the equations of motion for the mapping formalism [Eqs. (89a) and (89b)], the mean-field equations of motion for both electronic and nuclear DoF can thus be written in canonical form. 

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. 

Recall that the canonical equations of classical mechanics can be used to derive the Hamiltonian expression for the total energy of a system from the momenta pk and positional coordinates qk ... 

These equations are known as the canonical equations of motion. We also have ... 

The equations of motion (6.1.8) supplemented with a perfectly elastic refiection condition at x = 0 can be derived as the canonical equations of motion from the Hamiltonian... 

It is within the Hamiltonian formulation of classical mechanics that one introduces the concept of a canonical transformation. This is a transformation from some initial set of ps and qs, which satisfy the canonical equations of motion for H(p, q, t) as given in eqn (8.57), to a new set Q and P, which depend upon both the old coordinates and momenta with defining equations. 

Many authors have given analytic solutions with differing degrees of accuracy. Deslouis et al. developed a method that, after an approximation, reduces the problem to the canonical equation for Airy functions. Tribollet and Newman gave a solution under the form of two series one for K < 10 and one for K > 10. The two series overlapped well. 

The canonical equations of motion for system (1) can be cast in a form that uses the mean field (2) as follows ... 

Chose constraints for the system and apply these to limit the solutions to the canonical equations. Constraint conventions are used that limit the choices of how a particular problem can be solved. The problem is two-way constrained ... 

From a practical point of view, we shall never be able to perform that whole normalization process for a generic Hamiltonian. However, we can perform a finite number, r say, of steps, and consider the Hamiltonian H r) truncated at the column r of the diagram above as the approximate normal form that we are interested in. Let us call H r p r q ) the truncated Hamiltonian. Then the canonical equations for H r p r q ) admit the simple solution... 

The dynamics of the valve-positioning system may be described in terms of its small-signal and large-signal characteristics. The small-signal characteristics may be represented to reasonable accuracy by a first-order exponential lag, which will obey the canonical equation ... 

Before dealing with the general theory of the integration of the canonical equations, we will, first of all, consider some simple cases If the Hamiltonian function H does not contain one co-ordinate, e.g qlt i.e. if... 

In this case the canonical equations admit of immediate integration. We have... 

The problem of solving the system of 2f ordinary differential equations of the first order, i.e. the canonical equations, is therefore equivalent to that of finding a complete solution of the partial differential equation (3) (/ being greater than 1). This is a special case of general theorems on the relation between ordinary and partial differential equations. 

In this expression the wjs do not appear the Ja° s are therefore constant during the perturbed motion, and appear as parameters only the only variables arc the wp° s and Jp° s. These satisfy the canonical equations ... 

We now introduce the angle and action variables wk°, 3k° of the undisturbed motion, and consider the Hamiltonian function of the perturbed motion defined in terms of them. They are still canonical co-ordinates, but, in general, they are no longer angle and action variables in fact, it is evident from the canonical equations... 

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