Holonomic constraints

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In a system of particles, holonomic constraints can be expressed in the following form: f(q1, q2, q3,....,t) = 0 q1, q2, etc., are the coordinates of the particles. For example, the motion of a particle constraint to lie on the surface of a sphere is subject to a holonomic constraint, but if the particle is able to fall off the sphere under the influence of gravity than the constraint becomes non-holonomic.

Holonomic constraints are use in molecular dynamic simulations of molecules such as water to conserve hydrogens attached to an oxygen atom; thus, allowing a larger time step to be used. This simulations are performed classical methods. Since the length of a simulation time step used to integrate the trajectories of particles depends on the frequency of the fastest vibrating oscillators, that is a hydrogen atoms attached to a heavy atom, the time step would have to be too small to avoid the hydrogens to become unstable.If the hydrogen distance to the heavy atom is modeled as a holonomic constraint, the the simulation time step can be increased. The SHAKE algorithm implements this holonomic constraints by propagating the trajectory of hydrogens while still keeping the hydrogens with a fixed distance of the heavy atom.