holonomicity

Holonomicity is a concept found in various fields of mathematics and physics, most notably in differential geometry and classical mechanics. It generally refers to the property of a system or a set of constraints being "integrable" or "fully determined." In simpler terms, a holonomic system is one where the possible configurations can be described by a set of independent variables, and the relationships between these variables are fixed and can be integrated to find an overall description of the system's state.

In classical mechanics, holonomic constraints are restrictions on the motion of a system that can be expressed as algebraic equations involving the generalized coordinates and possibly time. These constraints reduce the number of degrees of freedom of the system. For example, if a bead is constrained to move along a wire, the position of the bead is restricted to the path of the wire, which can be described by an equation. If the constraint equation does not involve velocities or is velocity-independent and can be integrated, it is considered holonomic.

Conversely, non-holonomic constraints cannot be expressed in such a way. These constraints typically involve velocities or are differential relationships that cannot be integrated to yield a purely coordinate-dependent relationship. An example is a rolling wheel on a surface without slipping. The constraint relates the angular velocity of the wheel to the linear velocity of its center, and this relationship cannot be integrated to give a direct link between just the position and orientation. The study of holonomicity is crucial for understanding the configuration space of a system and for applying various analytical mechanics techniques.