Force constraint

Tig. 7.8 A box sliding down a slope under the influence of gravity is subject to the constraint that it must remain or the slope. The constraint force C acts perpendicular to the direction of motion. 

CE uses holonomic constraints. In a constrained system the coordinates of the particles 5t independent and the equations of motion in each of the coordinate directions are cted. A second difficulty is that the magnitude of the constraint forces is unknown, in the case of the box on the slope, the gravitational force acting on the box is in the ction whereas the motion is down the slope. The motion is thus not in the same direc-s the gravitational force. As such, the total force on the box can be considered to arise wo sources one due to gravity and the other a constraint force that is perpendicular to otion of the box (Figure 7.8). As there is no motion perpendicular to the surface of the the constraint force does no work. 

The constraint force can be introduced into Newton s equations as a Lagrange multipli (see Section 1.10.5). To achieve consistency with the usual Lagrangian notation, we wri F y as —A and so F Ar equals Am. Thus ... 

In deoxyhemoglobin, histidine F8 is liganded to the heme iron ion, but steric constraints force the Fe His-N bond to be tilted about 8° from the perpendicular to the plane of the heme. Steric repulsion between histidine F8 and the nitrogen atoms of the porphyrin ring system, combined with electrostatic repulsions between the electrons of Fe and the porphyrin 77-electrons, forces the iron atom to lie out of the porphyrin plane by about 0.06 nm. Changes in... 

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. 

Hopefully, we make our point. The confidence curve is NOT fashioned out of granite - it has to be applied with understanding and circumspection. It will always have the same basic shape but we have to be prepared to take a view on how wide it should be in every individual situation Matters become a great deal more complex when we come up against structures that are sterically crowded (i.e., structures where bond constraints force various moieties into close proximity with one another). 

Several techniques are available to calculate dJf /dQ. Ciccotti and coworkers [17, 19-27] have developed a technique, called blue-moon ensemble method or the method of constraints, in which a simulation is performed with fixed at some value. This can be realized by applying an external force, the constraint force, which prevents from changing. From the statistics of this constraint force it is possible to... 

Fig. 4.2. Free energy computation using constraint forces. It may be difficult to sample the surface (x) = using a constrained simulation because of the presence of energy barriers separating different reaction pathways. Left a barrier is shown in the middle of the pathway from reactant A to product B. Right two barriers are shown at B...

Before we derive the appropriate expressions to calculate cL4/d from constrained simulations, we note an important difference between sampling in constrained and unconstrained simulations. There are two ways to gather statistics at (x) = . In unconstrained simulations, the positions are sampled according to exp —iiU while the momenta are sampled according to exp —j3K. If a constraint force is applied to keep fixed the positions are sampled according to A( (x) — x) exp —iiU. The momenta, however, are sampled according to a more complex statistical ensemble. Recall that... 

The steepest descent (4.48) requires calculation of dA/d . Laio and Parrinello chose to estimate this term by performing short constrained molecular dynamics simulations and computing the potential of mean constraint force (see Carter et al. [45]). 

As the size of a polymerization batch is normalized (the size of a batch is 1 ) and the initial product amounts are defined to be zero (see below), the number of polymerizations N is a nonbinding upper bound for the product amounts. The first inequality constraint forces p to zero if the difference Znm — or if... 

It may be legacy in the sense that there is a lot of code and data already built for a given application or only in the sense that compatibility constraints force the use of more-traditional technology such as RDBs. 

The unimodality constraint distorts the least-squares improvements and this is evident from the slower convergence. However, the constraint forces the concentration profiles to physically possible shapes. 

The first of these expresses the condition that the centripetal constraint force does no work, because the velocity is perpendicular to the radius. The second states that the radial component of the acceleration is directed inwards and equal to the square of the speed. This relation can be used to calculate the constraint force by taking the scalar product of the equation of motion (2) with the vector function r[t], and using the constraint and its time derivatives to obtain... 

The preceding division of the hard components of the total nonhydrodynamic force F n(,x, in Eq. (2.121) between an elastic force F and a sum of constraint forces n(jX, is a matter of convention. Arbitrary shifts in the values of hard components of F could be absorbed into compensating shifts in the... 

More generally, contracting the transposed projection tensor with a force to its right (or the projection tensor with a force to its left) produces a constrained force given by the sum of the original force and the constraint force induced by it. Such constrained forces may have nonzero hard components, but, on contraction with induce velocities that do not. 

The reciprocal basis vector m) may be used to express Eq. (2.131) for the constraint forces more compactly, as... 

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