Constrained coordinates

LSD and Psilocybin I have been unable to identify any sign at all of addiction, organic injury, or other, in some way unpleasant aftereffects. .. In my opinion, however, LSD and psilocybin cannot and should not become "pleasure drugs" for the general public Their effects are such that they lead one beyond the customary (and constraining) coordinate system of space and time, and afford insights into the heaven and hell of one s own self — which can be dangerous to one who is not cut out for that, and hence is not prepared. 

The second approach, which uses the Lagrange multiplier technique, consists of retaining the set of constrained coordinates and making use instead of Lagrange s equations of the second kind, which involve the forces of constraints. The Lagrange equations of the second kind together with the equations of constraints are used to solve for both the coordinates and the forces of constraints. Use of this approach with Cartesian coordinates has come to be known as constraint dynamics. This chapter is concerned with the various methods of constraint dynamics. 

The integration of the equations of motion In this step, the forces of constraint and their time derivatives up to order s nax obtained from step 1, are used as input to the selected integration scheme, to generate the constrained coordinates. Note that the particular choice of numerical integration algorithm in step 2 determines the parameter of step 1. 

Equation [40] is used to Taylor-expand every holonomic constraint in the molecular model, in Eq. [39], about the partially constrained coordinates ]r (t + 8tj ]X<° (tg),. . . , ( " (tQ)])]. Standard tests must be used to establish... 

Comparing Eq. [51] with Eq. [52] leads to the identical expression, Eq. [50], for the approximate forces of constraints. Note the factor of in front of the last term in Eq. [52]. A common error in the literature is to extract the constraint forces Gjitg) from the basic Verier expression for the unconstrained coordinates, Eq. [48], and to write the constrained coordinates as follows ... 

Not surprisingly, comparing Eq. [53] with Eq. [51] leads to the forces of constraint in Eq. [50], but with a factor of V2 discrepancy. This discrepancy is due to unwarranted attempts to apply Eq. [48], which should be used only in the computation of the unconstrained coordinates [r (t(j + 8t)J, to the constrained coordinates [r(rQ + 8, (7))). Unfortunately, a factor of Va is often artificially introduced into the equations to mask this inconsistency. For convenience and conformity with the most widely adopted convention, the rest of this chapter redefines the undetermined parameters 7) such that their new values are equal to half their previously defined values. With this new definition, Eq. [46] takes the form... 

The constrained coordinates are obtained by means of Eq. [81]. The process is then repeated at the next time step. 

V X,l X),x) is evaluated directly from the MD trajectory or MC ensemble. In order to evaluate K(A + 5A,/(A-F 5A), a ), we first modify the constrained coordinates so that they have the appropriate values for state (A + SX) - extend or contract bonds, rotate torsions, etc. - and then calculate the potential energy from the resulting set of coordinates. 

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