Methods for Constrained Dynamics

An alternative, simpler, and probably more robust, procedure is simply to fix the value of A(q) by constraining q to its mechanical equilibrium value [189] in which case no fast averaging whatsoever needs to be performed. This approach relies on the use of the SHAKE method of constrained molecular dynamics (see the following section). 

In each of the mollified multiple timestepping methods, since only the potential energy is modified, and by a smooth mapping of positions, the forces derived by differentiating U are still conservative, and, just as with the MTS method, the scheme remains symplectic. 

Constraints are used frequently in molecular modelling. By a constraint, we mean a modification of a dynamical system to maintain the constancy of a given function of coordinates (or coordinates and momenta). In this section we consider holonomic constraints which may be expressed in terms of positions only, and perhaps time. 

Note that by differentiating (4.3) with respect to time, we obtain qg(q, t) q + dg/dt = 0 which involves velocities. The definition of holonomic constraints thus includes equations which relate positions and momenta, but only where these are, through the differential equations, equivalent to equations involving positions only (4.3). For more details on constrained dynamics, see for example [67, 79, 83, 164,227, 249], 

An important type of holonomic constraint arising in molecular dynamics relates to bond vibration. Recall that the stretch of a chemical bond between two atoms gives rise to a potential energy contribution of the form 

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AP93] Ascher U. M. and Petzold L. R. (1993) Stability of computational methods for constrained dynamic systems. SIAM J.Sci.Comp. 14 95-120. 

M. Sofer H. Brauchli, ODE Methods for Constrained Dynamics (in preparation). 

In this chapter we have presented a multi-scale method for molecular dynamics simulations of shock compression and characterized its behaviour. This method attempts to constrain the molecular dynamics system to the sequence of thermodynamic states that occur in a shock wave. While we have presented one particular approach, it is certainly not unique and there are likely a variety of related approaches to multi-scale simulations that have a variety of differing practical properties. These methods open the door to simulations of shock propagation on the longest timescales accessible by molecular d5nnamics and the use of accurate but computationally costly material descriptions like density fimctional theory. It is our belief that this method promises to be a valuable tool for elucidation of new science in shocked condensed matter. 

The method presented in the next section is an attempt to overcome the barrier due to the highest frequencies whatever their origin. Although it has been implemented and tested for unconstrained dynamics only, there is no fundamental reason why it cannot be applied to overcome the less restrictive time step barrier arising in constrained dynamics. 

Abstract. We present novel time integration schemes for Newtonian dynamics whose fastest oscillations are nearly harmonic, for constrained Newtonian dynamics including the Car-Parrinello equations of ab initio molecular dynamics, and for mixed quantum-classical molecular dynamics. The methods attain favorable properties by using matrix-function vector products which are computed via Lanczos method. This permits to take longer time steps than in standard integrators. 

The most commonly used method for applying constraints, particularly in molecula dynamics, is the SHAKE procedure of Ryckaert, Ciccotti and Berendsen [Ryckaert et a 1977]. In constraint dynamics the equations of motion are solved while simultaneous satisfying the imposed constraints. Constrained systems have been much studied in classics mechanics we shall illustrate the general principles using a simple system comprising a bo sliding down a frictionless slope in two dimensions (Figure 7.8). The box is constrained t remain on the slope and so the box s x and y coordinates must always satisfy the equatio of the slope (which we shall write as y = + c). If the slope were not present then the bo... 

Although constrained dynamics is usually discussed in the context of the geometrically constrained system described above, the same techniques can have many other applications. For instance, constant-pressure and constant-temperature dynamics can be imposed by using constraint methods [33,34]. Car and Parrinello [35] describe the use of the extended Lagrangian to maintain constraints in the context of their ab initio MD method. (For more details on the Car-Parrinello method, refer to the excellent review by Gain and Pasquarrello [36].)... 

Constraint dynamics is just what it appears to be the equations of motion of the molecules are altered so that their motions are constrained to follow trajectories modified to mclude a constraint or constraints such as constant (total) kinetic energy or constant pressure, where the pressure in a dense adsorbed phase is given by the virial theorem. In statistical mechanics where large numbers of particles are involved, constraints are added by using the method of undetermined multipliers. (This approach to constrained dynamics was presented many years ago for mechanical systems by Gauss.) Suppose one has a constraint g(R, V)=0 that depends upon all the coordinates R=rj,r2...rN and velocities V=Vi,V2,...vn of all N particles in the system. By differentiation with respect to time, this constraint can be rewritten as l dV/dt -i- s = 0 where I and s are functions of R and V only. Gauss principle states that the constrained equations of motion can be written as ... 

J. Bowman, B. Gazdy, and Q. Sun, A. method to constrain vibrational energy in quasiclass-ical trajectory calculations, J. Chem. Phys. 91 2859 (1989) R. Alimi, A. Garcia-Vela, and R. B. Gerber, A remedy for zero-point energy problems in classical trajectories a combined semiclassical/classical molecular dynamics algorithm, J. Chem. Phys. 96 2034 (1992). 

The SHAKE method has been extended by Tobias and Brooks [Tobias and Brooks 1988] to enable constraints to be applied to arbitrary internal coordinates. This enables the torsion angle of a rotatable bond to be constrained to a particular value during a molecular dynamics simulation, which is particularly useful when used in conjunction with methods for calculating free energies (see Section 11.7)... 

The physical effects of introducing constraints into a molecular model have been discussed by several authors. > This chapter is concerned mainly with the methods of constraint dynamics. In addition to descriptions of bond-stretch and angle-bend constraints, dihedral (or torsional) constraints are explicitly considered. Torsional modes generally have frequencies comparable to those of other modes, and the weak coupling condition is not satisfied in this case. Hence the constraint approximation is not justified for torsion. This fact is particularly important because torsional motions play a major role in conformation interconversion in small molecules as well as polymers, and constraining them can seriously alter the dynamics of the original, unconstrained system. 

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 analytical method deserves a detailed discussion for at least two major reasons. First, if used in conjunction with some constraint correction scheme," it is important in its own right as a practical method of solution of the constrained dynamics problem. Second, the method of undetermined parameters, central to the subject matter of this chapter, is an outgrowth of the analytical method hence a thorough understanding of the analytical method is an essential prerequisite for understanding the method of undetermined parameters. 

Actually, many methods in common use for molecular dynamics cannot be seen as maps of T M since the hidden constraint is allowed to be violated. A simple example of such a method is the constrained Symplectic Euler-Uke method... 

Equations 4.S9,4.60,4.67, and 4.70 summarize the fundamental and relevant recursive dynamic equations fw a constrained single chain. These equations wiU now be used to derive the Force Prqtagadon Method for computing the inverse operational space inertia matrix of a single chain. 

As mentioned earlier, the developed algorithm employs dynopt to solve the intermediate problems associated with the local interaction of the agents. Specifically, dynopt is a set of MATLAB functions that use the orthogonal collocation on finite elements method for the determination of optimal control trajectories. As inputs, this toolbox requires the dynamic process model, the objective function to be minimized, and the set of equality and inequality constraints. The dynamic model here is described by the set of ordinary differential equations and differential algebraic equations that represent the fermentation process model. For the purpose of optimization, the MATLAB Optimization Toolbox, particularly the constrained nonlinear rninimization routine fmincon [29], is employed. 

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