Holonomic constraints

In classical mechanics, holonomic constraints are relations between the position variables (and possibly time)^[1] that can be expressed in the following form:

$${\displaystyle f(u_{1},u_{2},u_{3},\ldots ,u_{n},t)=0}$$

where ${\displaystyle \{u_{1},u_{2},u_{3},\ldots ,u_{n}\}}$ are n generalized coordinates that describe the system (in unconstrained configuration space). For example, the motion of a particle constrained 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, the constraint becomes non-holonomic. For the first case, the holonomic constraint may be given by the equation

$${\displaystyle r^{2}-a^{2}=0}$$

where ${\displaystyle r}$ is the distance from the centre of a sphere of radius ${\displaystyle a}$, whereas the second non-holonomic case may be given by

$${\displaystyle r^{2}-a^{2}\geq 0}$$

Velocity-dependent constraints (also called semi-holonomic constraints)^[2] such as

$${\displaystyle f(u_{1},u_{2},\ldots ,u_{n},{\dot {u}}_{1},{\dot {u}}_{2},\ldots ,{\dot {u}}_{n},t)=0}$$

are not usually holonomic.^{[citation needed]}