Hamilton s equations of motion

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 ... 

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... 

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. 

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

Relation (6.40) represents a dynamical mapping which is mediated by Hamilton s equations of motion in the upper state and ultimately by the forces —dV/dR and —dV/dr and the torque —dV/d y. The energy dependence is mainly determined by the slope dV/dR of the potential in the direction of the dissociation path while dV/dr and dV/d y control the vibrational and rotational state distribution of the fragment. 

These equations are called Hamilton s equations of motion. It is easy to show that these various forms of Newton s equation of motion are equivalent. 

The integration over Q has been carried out, which implies that HAB sol is evaluated at Q = 0. From the Hamiltonian in Eq. (10.27), we find, using Hamilton s equations of motion,... 

Fig. A.2.1 An illustration of the 2s-dimensional phase space of a system with s degrees of freedom. The solid path describes the motion of the system according to Hamilton s equations of motion. The cube illustrates a phase-space cell of volume ha that contains one state.

In classical mechanics, the state variables change with time according to Hamilton s equations of motion... 

The first and second of these equations are Hamilton s equations of motion. The first and third establish wk = qk and pk = as dynamical conditions, equivalent... 

Then Hamilton s equations of motion are first-order differential equations ... 

PROBLEM 2.6.1. Solve the simple harmonic motion problem by using (i) Lagrange s and (ii) Hamilton s equations of motion. 

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,... 

Wiggins et al. [22] pointed out that one can always locally transform a Hamiltonian to the form of Eq. (1.38) if there exists a certain type of saddle point. Examination of the associated Hamilton s equations of motion shows that q = Pn = 0 is a fixed point that defines an invariant manifold of dimension 2n — 2. This manifold intersects with the energy surface, creating a (2m — 3)-dimensional invariant manifold. The latter invariant manifold of dimension 2m — 3 is an excellent example of an NHIM. More interesting, in this case the stable and unstable manifolds of the NHIM, denoted by W and W ,... 

Whether one uses Newton s or Hamilton s equations of motion, obtaining the atomic positions over time requires numerical integration. Integration of ordinary differential equations (ODE) is a well-traveled territory in numerical analysis. A number of different techniques are routinely used in MD. 

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