Canonical transformation theory defined

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... 

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... 

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. ... 

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. 

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