Hamilton’s equation

Figure Al.2.7. Trajectory of two coupled stretches, obtained by integrating Hamilton s equations for motion on a PES for the two modes. The system has stable anhamionic synmretric and antisyimnetric stretch modes, like those illustrated in figrne Al.2.6. In this trajectory, semiclassically there is one quantum of energy in each mode, so the trajectory corresponds to a combination state with quantum numbers nj = [1, 1]. The woven pattern shows that the trajectory is regular rather than chaotic, corresponding to motion in phase space on an invariant torus.

To generalize what we have just done to reactive and inelastic scattering, one needs to calculate numerically integrated trajectories for motions in many degrees of freedom. This is most convenient to develop in space-fixed Cartesian coordinates. In this case, the classical equations of motion (Hamilton s equations) are given... 

To develop an additional equation, we simply make the ansatz that the first temi on the left-hand side of equation (3.11.215h) equals the first temi on the right-hand side and similarly with the second temi. This innnediately gives us Hamilton s equations... 

The method of molecular dynamics (MD), described earlier in this book, is a powerful approach for simulating the dynamics and predicting the rates of chemical reactions. In the MD approach most commonly used, the potential of interaction is specified between atoms participating in the reaction, and the time evolution of their positions is obtained by solving Hamilton s equations for the classical motions of the nuclei. Because MD simulations of etching reactions must include a significant number of atoms from the substrate as well as the gaseous etchant species, the calculations become computationally intensive, and the time scale of the simulation is limited to the... 

As is well known. Molecular Dynamics is used to simulate the motions in many-body systems. In a typical MD simulation one first starts with an initial state of an N particle system F = xi,..., Xf,pi,..., pf) where / = 3N is the number of degrees of freedom in the system. After sampling the initial state one numerically solves Hamilton s equations of motion ... 

Although I do not intend to progress the idea here, there is a set of first-order differential equations called Hamilton s equations of motion that are fully equivalent to Newton s laws. Hamilton s equations are ... 

It is now shown how the abrupt changes in the eigenvalue distribution around the central critical point relate to changes in the classical mechanics, bearing in mind that the analog of quantization in classical mechanics is a transformation of the Hamiltonian from a representation in the variables pR, p, R, 0) to one in angle-action variables (/, /e, Qr, 0) such that the transformed Hamiltonian depends only on the actions 1r, /e) [37]. Hamilton s equations diR/dt = (0///00 j), etc.) then show that the actions are constants of the motion, which are related to the quantum numbers by the Bohr correspondence principle [23]. In the present case,... 

Perhaps the best starting point in a review of the nonequilibrium field, and certainly the work that most directly influenced the present theory, is Onsager s celebrated 1931 paper on the reciprocal relations [10]. This showed that the symmetry of the linear hydrodynamic transport matrix was a consequence of the time reversibility of Hamilton s equations of motion. This is an early example of the overlap between macroscopic thermodynamics and microscopic statistical mechanics. The consequences of time reversibility play an essential role in the present nonequilibrium theory, and in various fluctuation and work theorems to be discussed shortly. 

Evans and Baranyai [51, 52] have explored what they describe as a nonlinear generalization of Prigogine s principle of minimum entropy production. In their theory the rate of (first) entropy production is equated to the rate of phase space compression. Since phase space is incompressible under Hamilton s equations of motion, which all real systems obey, the compression of phase space that occurs in nonequilibrium molecular dynamics (NEMD) simulations is purely an artifact of the non-Hamiltonian equations of motion that arise in implementing the Evans-Hoover thermostat [53, 54]. (See Section VIIIC for a critical discussion of the NEMD method.) While the NEMD method is a valid simulation approach in the linear regime, the phase space compression induced by the thermostat awaits physical interpretation even if it does turn out to be related to the rate of first entropy production, then the hurdle posed by Question (3) remains to be surmounted. 

Perhaps the most common computer simulation method for nonequilibrium systems is the nonequilibrium molecular dynamics (NEMD) method [53, 88]. This typically consists of Hamilton s equations of motion augmented with an artificial force designed to mimic particular nonequilibrium fluxes, and a constraint force or thermostat designed to keep the kinetic energy or temperature constant. Here is given a brief derivation and critique of the main elements of that method. 

Fet us consider a system described by an explicitly time-dependent Hamiltonian p, q.t) where (p, q) = z is a point in phase space. Hamilton s equation of motion are... 

From Hamilton s equations, we find that dp/dp = —dq/dq, such that... 

From the definition of the work, (5.15), and Hamilton s equations, (5.17), it follows that the work performed along the trajectory is... 

For given initial conditions, trajectories in phase space satisfying Hamilton s equation of motion, (5.17), are given by... 

Consider a molecular system consisting of N atoms with Hamiltonian p) = K(p) I V (q). K(v) and V (q) are the kinetic and potential energy, respectively. Since the system evolves according to Hamilton s equations of motion... 

Problems in statistical mechanics of classical systems are nearly always treated by means of Hamilton s equations of motion. 

To show [115] that Liouville s theorem holds in any number of phase-space dimensions it is useful to restate some special features of Hamilton s equations,... 

The Bohr theory also permits a calculation of the total energy, E, of the atom from Hamilton s equation ... 

In principle, with given initial conditions, the time evolution of the coordinates and momenta [rjt p ) of each particle is given by Hamilton s equations 15... 

Taking into account that the particles of the system all move according to Hamilton s equations (4), it is easily shown that the distribution function at later times, t > 0, is determined by the Liouville equation11 >40... 

The Hamilton s equations for three-body system described by the general dynamical coordinates qp are... 

If we want to study the implications of various features of potential energy surface to dynamical results we have to carry out the dynamics. From a practical point of view, we can use classical mechanics. One numerically solves Hamilton s equations... 

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. 

A major preoccupation in nonequilibrium statistical mechanics is to derive hydrodynamics and nonequilibrium thermodynamics from the microscopic Hamiltonian dynamics of the particles composing matter. The positions raYl= and momenta PaY i= of these particles obey Newton s equations or, equivalently, Hamilton s equations ... 

Since the Hamiltonian function // is an even function of the momenta, Hamilton s equation are symmetric under time reversal ... 

This solid line is obtained by numerical integration of Hamilton s equations of motion for the original variables. 

Constraints may be introduced either into the classical mechanical equations of motion (i.e., Newton s or Hamilton s equations, or the corresponding inertial Langevin equations), which attempt to resolve the ballistic motion observed over short time scales, or into a theory of Brownian motion, which describes only the diffusive motion observed over longer time scales. We focus here on the latter case, in which constraints are introduced directly into the theory of Brownian motion, as described by either a diffusion equation or an inertialess stochastic differential equation. Although the analysis given here is phrased in quite general terms, it is motivated primarily by the use of constrained mechanical models to describe the dynamics of polymers in solution, for which the slowest internal motions are accurately described by a purely diffusive dynamical model. 

To evaluate the average mechanical force (pa), we combine Hamilton s equation for pa, Eq. (2.81), with the preceding assumptions to obtain... 

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