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In computational chemistry, a constraint algorithm is a method for satisfying a Newtonian motion of a rigid which consists of mass points. By this algorithm, the distance between mass points is maintained constant. In general, the algorithm is constructed by following procedures; (i) choosing novel unconstrained coordinates (internal coordinates), (ii) introducing explicit constraint forces, (iii) minimizing constraint forces implicitly by the technique of Lagrange multipliers or projection methods. Constraint algorithms are often applied to molecular dynamics simulations. Although such simulations are sometimes performed using internal coordinates that automatically satisfy the bond-length, bond-angle and torsion-angle constraints, simulations may also be performed using explicit or implicit constraint forces for these three constraints. However, explicit constraint forces give a cause of the inefficient computation; more computational power is required to get a trajectory of a given length. Therefore, internal coordinates and implicit-force constraint solvers are generally preferred. Constraint algorithms achieve efficient computational time by neglecting motions of intramolecular atoms. If intramolecular behavior is important, e.g. hydrogen bonding and its behavior, constraint algorithms should not be used. ==Mathematical background== The motion of a set of ''N'' particles can be described by a set of second-order ordinary differential equations, Newton's second law, which can be written in matrix form : where the index ''j'' runs from 1 to ''M''. For brevity, these functions ''g''''i'' are grouped into an ''M''-dimensional vector g below. The task is to solve the combined set of differential-algebraic (DAE) equations, instead of just the ordinary differential equations (ODE) of Newton's second law. This problem was studied in detail by Joseph Louis Lagrange, who laid out most of the methods for solving it. The simplest approach is to define new generalized coordinates that are unconstrained; this approach eliminates the algebraic equations and reduces the problem once again to solving an ordinary differential equation. Such an approach is used, for example, in describing the motion of a rigid body; the position and orientation of a rigid body can be described by six independent, unconstrained coordinates, rather than describing the positions of the particles that make it up and the constraints among them that maintain their relative distances. The drawback of this approach is that the equations may become unwieldy and complex; for example, the mass matrix M may become non-diagonal and depend on the generalized coordinates. A second approach is to introduce explicit forces that work to maintain the constraint; for example, one could introduce strong spring forces that enforce the distances among mass points within a "rigid" body. The two difficulties of this approach are that the constraints are not satisfied exactly, and the strong forces may require very short time-steps, making simulations inefficient computationally. A third approach is to use a method such as Lagrange multipliers or projection to the constraint manifold to determine the coordinate adjustments necessary to satisfy the constraints. Finally, there are various hybrid approaches in which different sets of constraints are satisfied by different methods, e.g., internal coordinates, explicit forces and implicit-force solutions. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Constraint algorithm」の詳細全文を読む スポンサード リンク
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