Holonomic constraints Subject

The concept of a symplectic method is easily extended to systems subject to holonomic constraints [22]. For example the RATTLE discretization is found to be a symplectic discretization. Since SHAKE is algebraically equiva lent to RATTLE, it, too, has the long-term stability of a symplectic method. 

This section presents the notation for generalized coordinates, constraints, basis vectors, and tensors that is used throughout the paper. We consider a system consisting of N pointlike particles (beads) with positions R, ..., R with masses mi,..., mj. The positions of the beads are subject to K holonomic constraints, of the form... 

Any conservative mechanical system which is either free or subject to holonomic constraints and whose potential does not depend on the generalized velocities is described by standard equations of motion (either Lagrangian or Hamiltonian). The kinetic energy of the iV-particIe system is ... 

We finally note that is possible to use Gauss principle to obtain equations of motion when the system is subject to holonomic constraints such as bond length or bond angle constraints. In this case one obtains the same equations of motion as one would obtain by applying the Lagrange equation. 

Although this chapter is concerned primarily with holonomic constraints, we comment on the role of nonholonomic constraints in the present context. Because the analytical dynamics theory of a system of particles subject to holonomic constraints is well established, the issues in (holonomic) constraint dynamics are mainly algorithmic in nature, as seen in this chapter. The situation is more complex for molecular dynamics with nonholonomic constraints, where theoretical difficulties exist. " ... 

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