Canonical transformation defined

Benettin, G., Galgani, L., Giorgilli, A. and Strelcyn, J. M. (1984). A proof of Kolmogorov s theorem on invariant tori using canonical transformations defined by the Lie method. R Nuovo Cimento, 79 201-216. 

The two pictures above (where U is viewed as acting on the wavefunction or acting on the Hamiltonian) are clearly mathematically equivalent. However, it is worth considering their physical equivalence in the language of canonical transformations. (A similar discussion of this issue may also be found in White [22].) In the first picture the Hamiltonian H, wavefunctions to and and transformation U are associated with particles defined by the operators c, Cj thus... 

In the current work, we consider primarily two theoretical models the linearized canonical transformation with doubles (L-CTD) and linearized canonical transformation with singles and doubles (L-CTSD) theories. These are defined by the choice of operators in A. The L-CTD theory contains only two-particle... 

Note that, in this case, the magnetic lines are contained in magnetic surfaces. There are in fact two families of them, given by the equations p = po and q = qo, where each line forms the intersection of two surfaces, one of each family (there are also two families of electric surfaces u = it0 and v = Vo). The functions (p, q) and ( , v) are the Clebsch variables of B and E, respectively [63,64], They can be used as canonical variables [62]. As explained above, they are not uniquely defined, but may be changed by canonical transformations. 

Further reduction of the constrained reaction path model is possible. Here we adopt a system-bath model in which the reaction path coordinate defines the system and all other coordinates constitute the bath. The use of this representation permits the elimination of the bath coordinates, which then increases the efficiency of calculation of the motion along the reaction coordinate. In particular. Miller showed that a canonical transformation of the reaction path Hamiltonian T + V) yields [38]... 

One of the main advantages of the Hamiltonian formalism is that it treats coordinates and momenta on an equal footing. In order to simpUfy the formulation and solution of mechanical problems it is often useful to define new coordinates and momenta which are mixtures of the old coordinates and momenta. But in order not to destroy the basic structure of the theory, only transformations which preserve the canonical structure (3.1.21) are allowed. These transformations are also called canonical transformations because they respect the canonical structme of the equations of motion (3.1.21). A transformation from the old momenta and coordinates p and q, to new momenta P and new coordinates Q according to... 

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. 

There remains but one important concept to complete our summary of the role of canonical transformations in classical mechanics, that of the Poisson bracket. Let F p,q) and G p,q) denote two mechanical properties of the system. Their Poisson bracket is defined as... 

An efficient treatment of the time evolution operator defined in Eq. (2.33) can be achieved by performing a canonical transformation of coordinates acting on the field X. We define the shifted vector X— X -p.,Uj — p.2 2 a new set of field coordinates. The potential is now decoupled... 

The SRTS sequence consists of a preparatory pulse and an arbitrary long train of the phase-coherent RF pulses of the same flip angle applied with a constant short-repetition time. As was noted above, the "short time" in this case should be interpreted as the pulse spacing T within the sequence that meets the condition T T2 Hd. The state that is established in the spin system after the time, T2, is traditionally defined as the "steady-state free precession" (SSFP), ° and includes two other states (or sub-states) quasi-stationary, that exists at times T2effective relaxation time) and stationary, that is established after the time " 3Tie after the start of the sequence.The SSFP is a very particular state which requires a specific mechanism for its description. This mechanism was devised in articles on the basis of the effective field concept and canonical transformations. Later approaches on the basis of the average-Hamiltonian theory were developed. ... 

Canonical transformations are used to reformulate the Hamiltonian of a system in new coordinates that render it more amenable to analysis [101]. The transformation of H is defined by a unitary operator, U, that acts to produce a new Hamiltonian, H = U HU. Because U is unitary, it is guaranteed to preserve the eigenvectors and eigenvalues of the original Hamiltonian. When H is written in terms of some set of variables (operators) A, then H will have the same form when written in terms of the transformed variables A = U AylJ, each of which is in general a function of all the untransformed variables [i.e., A. =/( Ay )]. To make progress, then, we use the latter relationship to write the transformed Hamiltonian H in terms of the untransformed variables, A (see Ref. 101). 

The radial variable r is dimensionalized to isolate the Damkohler number in the mass balance. It is important to emphasize that dimensional analysis on the radial coordinate must be performed after implementing the canonical transformation from Ca to iJia- If the surface area factors of and 1/r are written in terms of as defined by equation (13-9), prior to introducing the canonical transformation given by equation (13-4), then the mass transfer problem external to the spherical interface retains variable coefficients. If diffusion and chemical reaction are considered inside the gas bubble, then the order in which the canonical transformation and dimensional analysis are performed is unimportant. Hence,... 

The third equivalent formulation of classical mechanics to be briefly discussed here is the Hamiltonian formalism. Its main practical importance especially for molecular simulations lies in the solution of practical problems for processes that can be adequately described by classical mechanics despite their intrinsically quantum mechanical character (such as protein folding processes). However, more important for our purposes here is that it can serve as a useful starting point for the transition to quantum theory. The basic idea of the Hamiltonian formalism is to eliminate the / generalized velocities in favor of the canonical momenta defined by Eq. (2.54). This is achieved by a Legendre transformation of the Lagrangian with respect to the velocities. 

It is always defined with respect to the given set of canonical variables which is sometimes symbolized by suitable indices. The result, i.e., the Poisson bracket itself is, in general, another phase space function. In anticipation of the next section 2.3.3, all Poisson brackets are invariant under canonical transformations, i.e., their value does not depend on the choice of canonical variables. The indices q and p attached to the Poisson brackets are therefore often suppressed. 

Example 5.1 Transformation of Ordinary Differential Equations into Their Canonical Form Apply the transformations defined by Eqs. (5.31) and (5.32) to the following ordinary differential equations ... 

It is also possible to define real normal coordinates n(j) which describe running waves. In contrast to the real normal coordinates a ( ), the coordinates n( ) are not just linear combinations of Q(S) and Q ( ) but also of the momenta P(p and P (. They are coordinates which can be obtained by a canonical transformation, but we shall introduce them here in an elementary way as follows [2.6] ... 

This coordinate transformation gives rise to a corresponding transformation of the momenta via the canonical lift transformation [10]. Thus the corresponding conjugate momenta are p R, defined by... 

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 ... 

We define a canonical form for schemes as a class SP of schemes such that there is an algorithm which transforms any scheme into a strongly equivalent member of SP. Thus one has ... 

If there are n0 open channels at energy E, there are n linearly independent degenerate solutions of the Schrodinger equation. Each solution is characterized by a vector of coefficients aips, for i = 0,1, defined by the asymptotic form of the multichannel wave function in Eq. (8.1). The rectangular column matrix a consists of the two n0 x n0 coefficient matrices ao, < i Any nonsingular linear combination of the column vectors of a produces a physically equivalent set of solutions. When multiplied on the right by the inverse of the original matrix a0, the transformed a-matrix takes the canonical form... 

---