Hamiltons Equations of Motion

The total effective Hamiltonian H, in the presence of a vector potential for an A + B2 system is defined in Section II.B and the coupled first-order Hamilton equations of motion for all the coordinates are derived from the new effective Hamiltonian by the usual prescription [74], that is. 

To perform MD simulation of a system with a finite number of degrees of freedom the Hamilton equations of motion... 

We consider an ensemble of systems each containing n atoms. Thus, q = (<71, , < 3n), P = (pi, , P3n), and dpdq = Ilf" (dp dqi). We assume that all interactions are known. As time evolves, each point will trace out a trajectory that will be independent of the trajectories of the other systems, since they represent isolated systems with no coupling between them. Since the Hamilton equations of motion, Eq. (4.63), determine the trajectory of each system point in phase space, they must also determine the density p(p, q, t) at any time t if the dependence of p on p and q is known at some initial time to. This trajectory is given by the Liouville equation of motion that is derived below. 

A q) exists moreover, it is real and symmetric. It is important to note that obtaining T (q, p) is no more difficult than an inversion of A (q). The Hamilton equations of motion of the second type then are ... 

A suitably chosen set of continuous parameters ft can be viewed as generalized coordinates that are propagated in time by the Hamilton equations of motion [52]... 

In the special case that the generalized coordinates ft represent the Cartesian coordinates of n point masses and, furthermore, that momenta can be separated from coordinates in the Hamiltonian H, the Hamilton equations of motion reduce to the more familiar Newton s second law ... 

To derive the Lagrange and Hamilton equations of motion we need expressions for the kinetic and potential energies ... 

The Ml manifestation of chaotic behavior requires that at least Mee equations of motion are coupled. This is not a strong requirement for a mechanical system. Each degree of freedom gives rise to two (Hamilton) equations of motion (or one, but second-order, Newton equation). So two coupled (anharmonic) oscillators can already exhibit chaotic behavior. Solving the trajectory for an atom colliding with a diatom requires six equations. If there are only two variables, one can get oscillatory solutions but not chaos. 

Differentiating the Hamilton equations of motion for the characteristics with respect... 

In classical the Hamilton equations of motion are solved numerically... 

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