in General

If you have a mechanical system with 𝑁 particles, you’d technically need 𝑛=3𝑁 coordinates to describe it completely.

But often it is possible to express one coordinate in terms of others: for example of two points are connected by a rigid rod, their relative distance does not vary. Such a condition of the system can be expressed as an equation that involves only the spatial coordinates 𝑞𝑖 of the system and the time 𝑡, but not on momenta 𝑝𝑖 or higher derivatives wrt time. These are called holonomic constraints: 𝑓(𝑞𝑖,𝑡)=0. The cool thing about them is that they reduce the degrees of freedom of the system. If you have 𝑠sconstraints, you end up with 𝑛′=3𝑁−𝑠<𝑛 degrees of freedom.


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Non-holonomic constraints are basically just all other cases: when the constraints cannot be written as an equation between coordinates (but often as an inequality).

An example of a system with non-holonomic constraints is a particle trapped in a spherical shell. In three spatial dimensions, the particle then has 3 degrees of freedom. The constraint says that the distance of the particle from the center of the sphere is always less than 𝑅: √𝑥2+𝑦2+𝑧2<R. We cannot rewrite this to equality, so this is a non-holonomic, scleronomous constraint.

http://galileoandeinstein.physics.virginia.edu/7010/CM_29_Rolling_Sphere.pdf

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