What are holonomic constraints?

Question

I'm trying to understand what holomic constraints are mathematically. Wikpedia says:

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

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

where ${\displaystyle \{q_{1},q_{2},q_{3},\ldots ,q_{n}\}}$ are the $n$ coordinates which describe the system.

This denifition is very vague to me because unless you define $f$ I can't see the "form" they are talking about.

My question: I've a particle of mass, $m$, is constrained to move under gravity on the surface of a cone defined by the equation $x^2 +y^2 =z^2$ for $z \ge 0$. How would I determine/prove whether or not this is holomonic constraint?

Answer 1

If the particle is constrained to be alway on the cone, then you have holonomic constraints. If instead the particle simply cannot go inside the cone (but can "jump off"), then those are non-holonomic constraints.

Clearer examples:

• standard block sliding along an undulating track is non-holonomic. After all, if the track drops away too quickly, the block will fly off.
• a bead (with hole) sliding along a wire (through its middle) is holonomic. No matter how much the wire curves, the bead will always stay on.

At base, the difference is expressed mathematically as the difference between $=$ and $\geq$....

Answer 2

A holonomic constraint is an integrable constraint, or also in other words, offer restrictions to generalized positions. Ex. The constraint in the plane movement

$$ x_1 \dot x_1 + x_2\dot x_2 = 0 $$

This is a holonomic constraint because it comes from

$$ \frac{d}{dt}(x_1^2+x_2^2)=0\Rightarrow x_1^2+x_2^2=C $$

The constraint is non-holonomic when it can't be represented as a derivative regarding time from an integral expression, or in other words, offer restrictions between generalized positions and generalized velocities.

Ex. From a skate movement along a line with direction $\theta$

$$ \dot x_1 = v\cos\theta\\ \dot x_2 = v\sin\theta $$

after division we have

$$ \dot x_2 -\dot x_1\tan\theta=0 $$

This relationship cannot be integrated because $\theta$ is independent of $x_1, x_2$