Equation Hamilton

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... 

The explicit symplectic integrator can be derived in terms of free Lie algebra in which Hamilton equations (5) are written in the form... 

Therefore S can be constructed by solving the Lagrange equations for the nuclear variables. Alternatively it can be obtained from the Hamilton equations introducing the generalized momenta P = dS/dQ and solving the equations... 

With the new density matrices known up to time t, it is possible to advance the nuclear positions and momenta by integrating their Hamilton equations. This completes a cycle which can be repeated to advance to a later time t2-This sequence based on relaxing the density matrix for fixed nuclei and then correcting it to account for nuclear motions has been called the relax-and-drive procedure, and has been numerically implemented in several applications.(16, 15, 21, 46, 35-37)... 

A marker substance is injected into a central vein. A peripheral arterial line is used to measure the amount of the substance in the arterial system. A graph of concentration versus time is produced and patented algorithms based on the Stewart-Hamilton equation (below) are used to calculate the cardiac output. 

But the major physical problem remained open Could one prove rigorously that the systems studied before 1979—that is, typically, systems of N interacting particles (with N very large)—are intrinsically stochastic systems In order to go around the major difficulty, Prigogine will take as a starting point another property of dynamical systems integrability. A dynamical system defined as the solution of a system of differential equations (such as the Hamilton equations of classical dynamics) is said to be integrable if the initial value problem of these equations admits a unique analytical solution, weekly sensitive to the initial condition. Such systems are mechanically stable. In order to... 

The phase flow of the Hamilton equations preserves the Lagrange property. Thus, the manifold Lt at any t is a Lagrange manifold (even if the particles move not by inertia, but in any potential field). And so, Ya.B. s pancake theory is a theory of the caustics which form in the mapping of Lagrange manifolds from phase space onto configuration space. 

When the 0-0 bond breaks, the angular dependence of the upper-state PES generates a torque —dV/dp as discussed in Section 10.2.1. In analogy with (5.4f), the corresponding Hamilton equations for the evolution of ji and 32 are... 

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. 

From the Hamiltonian (1) with V(x) = 0, one deduces the Hamilton equations for all the degrees of freedom of the global system that is, for the particle, we have... 

Equations (6) and (10) have been deduced without approximation from the Hamilton equations (2) and (3). The memory kernel and the random force are expressed in terms of the parameters of the microscopic Caldeira-Leggett Hamiltonian (1). 

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 ... 

Here, a is an arbitrary constant. Now, we see that the dissipative part of the kinetic matrix satisfies the Onsager symmetry relation and the positivity of the damping constant c trivially. The constant a can be evaluated as follows. In the case of zero for the dissipative part of the kinetic matrix, these equations must be transformed into Hamilton equations of a simple harmonic oscillator. From this fact, it follows that a is a universal constant and a=1. Also, the final form of the Onsagerian constitutive (kinetic) equation the of damped oscillator is... 

The non-degenerate variables satisfy the same Hamilton equations of the Hamiltonian system considered in Figure 4, with perturbing parameter... 

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