Generalized constraint forces

The transformation to state space form reduces the number of diflFerential equations to its minimum by incorporating the constraint equations. In the resulting equations the generalized constraint force no longer appears and an additional equation must be provided if its value is required. 

The same procedure but with premultiplication by G results in an expression for the generalized constraint forces... 

This linear equation can be solved by standard methods, if G has maximal rank. Otherwise, the generalized constraint forces are no longer uniquely defined. 

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. 

More generally, m)" Fy is the constraint force induced by an unconstrained force Fy. 

General quartic force field with three quartic constraints. 

General quartic force field with three quartic constraints. c Tentative values based on a modified valence-force model with nine parameters. 

This together with (1.4.2) and (1.4.5) is the formulation of the equations of motion in a minimal set of coordinates. Constraint forces and constraints have been eliminated by this transformation. In case of tree structured systems this is a transformation to a state space form. The general way to transform the equations of motion into a state space form and a definition of these forms will be given in Ch. 2.1. 

The scaling prescription (59) embodies the assumption that the external force is so weak that it does not drive the TS trajectory out of the phase-space region in which the normal form expansion is valid. In the autonomous version of geometric TST, one generally assumes that this region is sufficiently large to make the normal form expansion a useful tool for the computation of the geometric objects. Once this assumption has been made, the additional condition imposed by Eq. (59) is only a weak constraint. 

In this chapter, we focus on the method of constraints and on ABF. Generalized coordinates are first described and some background material is provided to introduce the different free energy techniques properly. The central formula for practical calculations of the derivative of the free energy is given. Then the method of constraints and ABF are presented. A newly derived formula, which is simpler to implement in a molecular dynamics code, is given. A discussion of some alternative approaches (steered force molecular dynamics [35-37] and metadynamics [30-34]) is provided. Numerical examples illustrate some of the applications of these techniques. We finish with a discussion of parameterized Hamiltonian functions in the context of alchemical transformations. 

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