Holonomic constraints

In <a href="/facts/Classical_mechanics/ykG8upsN">classical mechanics</a>, holonomic constraints are relations between the position variables (and possibly time) that can be expressed in the following form:

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

where 
 
 
 
 {
 
 u
 
 1
 
 
 ,
 
 u
 
 2
 
 
 ,
 
 u
 
 3
 
 
 ,
 …
 ,
 
 u
 
 n
 
 
 }
 
 
 {\displaystyle \{u_{1},u_{2},u_{3},\ldots ,u_{n}\}}
 
 are n generalized coordinates that describe the system (in unconstrained <a href="/facts/Configuration_space_(physics)/LglVrWpq">configuration space</a>). For example, the motion of a particle constrained to lie on the surface of a <a href="/facts/Sphere/jywYLNM2">sphere</a> 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

 
 
 
 
 r
 
 2
 
 
 −
 
 a
 
 2
 
 
 ≥
 0
 
 
 {\displaystyle r^{2}-a^{2}\geq 0}

Velocity-dependent constraints (also called semi-holonomic constraints) such as

 
 
 
 f
 (
 
 u
 
 1
 
 
 ,
 
 u
 
 2
 
 
 ,
 …
 ,
 
 u
 
 n
 
 
 ,
 
 
 
 
 u
 ˙
 
 
 
 
 1
 
 
 ,
 
 
 
 
 u
 ˙
 
 
 
 
 2
 
 
 ,
 …
 ,
 
 
 
 
 u
 ˙
 
 
 
 
 n
 
 
 ,
 t
 )
 =
 0
 
 
 {\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.

Holonomic constraints open-in-new

Holonomic constraints