Hamilton canonical equations

The first step in the study of collision dynamics is to assume that nuclear motion obeys tha laws of classical mechanics. This approximation is expected to give, at least qualitatively, a correct description of the collision between heavy particles at high (relative) velocities. The most appropriate formalism for such a description is based on the Hamilton canonical equations of motion... 

For systems with no constraints, Newton s laws of motion are the same as the Hamilton canonical equations of motion obtained from the Hamiltonian function... 

We next establish the nuclear equations of motion. The equations of motion analogous to the Hamilton canonical equations of motion establishes the following set of equations. After some manipulation [423, 492, 493[, we have... 

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. 

There exists a special type of coordinate transformation in phase space, called a canonical transformation, which transforms the original system variables (q,p) to new system variables q, p ) = q, q 2, , q p, P2, , p ) while retaining the structure of Hamilton s equations of motion, that is,... 

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. 

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. 

Earlier in this section it was commented on how the minimal-coupling QED Hamiltonian is obtained from fhe classical Lagrangian function. A few words are in order regarding the derivation of the multipolar Hamiltonian (6). One method involves the application of a canonical transformation to the minimal-coupling Hamiltonian [32]. In classical mechanics, such a transformation renders the Poisson bracket and Hamilton s canonical equations of motion invariant. In quantum mechanics, a canonical transformation preserves both the commutator and Heisenberg s operator equation of motion. The appropriate generating function that converts H uit is propor-... 

We shall place ourselves in the Hamiltonian framework. We consider a 2n-dimensional phase space F endowed with canonical coordinates qi,..., qn,pi,..., pn. The flow in the phase space is determined by a smooth Hamiltonian function F[(q,p,t) via the Hamilton s equations... 

In this section, we are concerned with the canonical equations of the radiation field. We consider the fact that the electromagnetic wave is a transverse wave, and convert it into the form of Hamilton kinetic equations which are independent of the transformation parameter. In this process we will reach the conclusion that the radiation field is an ensemble of harmonic oscillators. During this process we will stress the concepts of vector potential and scalar potential. The equations of an electromagnetic wave in the vacuum are summarized as follows ... 

As shown in Appendix B, it can easily be proved for any transforms described by the functional form of Eq. (2.15), that if z(0) are canonical, z(s) are also canonical (and vice versa), as the time evolution of any Hamiltonian system is regarded as a canonical transformation from canonical variables at an initial time to those at another time, maintaining the structure of Hamilton s equations. 

Such a system generally does not have analytically integrable equations of motion. However, we may apply Hamilton s equations of motion, solve them numerically, and thus generate a unique trajectory for each set of initial conditions we choose. The resulting dynamics generally exhibits a variety of interesting phenomena. First, the frequency of motion in each mode is no longer a constant [as would be the case if we had f(q, qi) = 0] but depends on the instantaneous values of the canonical coordinates ( p, ) ... 

First we consider a system with two degrees of freedom (N = 2). Suppose we have two closed curves yi and y2 phase space, both of which encircle a tube of trajectories generated by Hamilton s equations of motion. These curves can be at two sequential times (tj, or they can be at two sequential mappings on a Poincare map These curves are associated with domains labeled (Dj, D2), which are the projections of the closed curves upon the coordinate planes (pj, qj. Because both the mapping and the time propagation are canonical transformations, the integral invariants (J-j,. 2) are preserved (constant) in either case. There are two of them, of the form ... 

Similarly, it can be shown that Eq. [55] is a statement of the preservation of phase space volume under propagation by Hamilton s equations of motion, that is, Liouville s theorem. It is important to note that the Poincare integral invariants are also preserved under a canonical transformation of any kind and not just the propagation of Hamilton s equations. 

More complicated is the treatment of the collision of an atom A with a diatomic molecule BC. In this case the canonical equations of motion (14.11) must be used for a system of three atoms A,B,C with 6 nuclear degrees of freedom (in a center-of-mass coordinate system). The Hamilton function for the initial configuration (large separation of atom A from molecule BC) is conveniently written in the form... 

Pi = Piipi,q, P2,92, , t). Those transformations that do retain the form of Hamilton s equations are said to be canonical transformations In practice, canonical transformations can often greatly simplify the solutions to Hamilton s equations. For more discussion on the use of canonical transformations in theoretical mechanics, please see the Further Reading section list at the end of the chapter. 

Finally, before leaving this section, we note another important aspect of the Liouville equation regarding transformation of phase space variables. We noted in Chap. 1 that Hamilton s equations of motion retain their form only for so-called canonical transformations. Consequently, the form of the Liouville equation given above is also invariant to only canonical transformations. Furthermore, it can be shown that the Jacobian for canonical transformations is unity, i.e., there is no expansion or contraction of a phase space volume element in going from one set of phase space coordinates to another. A simple example of a single particle in three dimensions can be used to effectively illustrate this point.l Considering, for example, two representations, viz., cartesian and spherical coordinates and their associated conjugate momenta, we have... 

Standard molecular dynamics calculations, i.e., those that solve Hamilton s equation, are performed on NVE ensembles, i.e., samples with a constant number of atoms N), fixed volume (V), and constant energy ( ). In standard Monte Carlo simulations the more widely applicable NVT ensembles are used, i.e., constant temperature (T) rather than energy, although both schemes can be modified to work in different ensembles. In particular, free energies can be directly evaluated using Monte Carlo methods in the Grand Canonical ensemble, although technical difficulties involved... 

Action-angle variables can also be introduced for certain types of motion in systems with many degrees of freedom, providing there exists one or more sets of coordinates in which the HJ equation is completely separable. If only conservative systems are considered Hamilton s characteristic function should be used. Complete separability means that the equations of canonical transformation have the form... 

It should be noted that the classical equations of motion (usually in the Hamilton s canonical form) are solved by various numerical methods and the... 

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