Holonomic constraints

In classical mechanics, holonomic constraints are relations between the position variables (and possibly time) which can be expressed in the following form: f ( q 1 , q 2 , q 3 , … , q n , t ) = 0 {displaystyle f(q_{1},q_{2},q_{3},ldots ,q_{n},t)=0} where { q 1 , q 2 , q 3 , … , q n } {displaystyle {q_{1},q_{2},q_{3},ldots ,q_{n}}} are the n coordinates which describe the system. 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: r 2 − a 2 = 0 {displaystyle r^{2}-a^{2}=0} Where r {displaystyle r} is the distance from the centre of a sphere of radius a {displaystyle a} . whereas the second non-holonomic case may be given by: