Canonical equation

An equation in which the entropy of a homogeneous fluid is expressed as a function of its energy and volume is called by Planck (1909) a canonical equation. 

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. 

When the response surface has an extreme, then all coefficients of a canonic equation have the same signs and the center of the figure is close to the center of experiment. A saddle-type surface has a canonic equation where all coefficients have different signs. In a crest-type surface some canonic equation coefficients are insignificant and the center of the figure is far away from the center of experiment. To obtain a surface approximated by a second-order model for two factors, it is possible to get four kinds of contour curves-graphs of constant values ... 

The theory is usually expressed in terms of canonical equations... 

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 ... 

Just as the fundamental property relation of Eq. (13.12) provides complete property information from a canonical equation of state expressing G/RT as a function of T, P, and composition, so the fundamental residual-property relation, Eq. (13.13) or (13.14), provides complete residual-property information from a PVT equation of state, from PVT data, or from generalized PVT correlations. However, for complete property information, one needs in addition to PVT data the ideal-gas-state ieat capacities of the species that comprise the system. 

The Hamiltonian function, together with its associated canonical equations of motion, can be derived in the following way. 

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

Suppose we investigate an autonomous Hamiltonian system with / degrees of freedom. Then, the solution of the 2/ first order canonical equations (3.1.21) contains 2/ integration constants Ck, fc = 1,..., 2/ according to... 

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... 

In ref. 139 the authors present an explicit method for the numerical solution of the Schrodinger equation. The Schrodinger equation is first transformed into a Hamiltonian canonical equation and then the author obtained several methods up to the... 

A pure fluid is described by tire canonical equation of state G = FfT) "I" RT hr P, where FfT) is a substarrce-specificfunctionof temperature. Detemrine for such a fluid expressions for V, S, H, U, Cp, and Cy. These results are consistent with those for an important model of gas-phase belravior. Wliat is the model ... 

These are Hamilton s canonical equations. One may determine the temporal behaviour of a classical system with N degrees of freedom by solving Lagrange s N second-order differential equations with the constants of integration being fixed by the IN initial values of the coordinates and Velocities which determine the initial state of the system, or by solving Hamilton s IN first-order equations for the same initial state. 

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. 

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