Bead on a rotating wire - holonomic or not?

[bitty]

Homework Statement

We have a bead sliding on a frictionless hoop oriented vertically. First the hoop rotates about its center with rotation axis perpendicular to its plane.

Second, the hoop rotates about a vertical axis as well.
In both of these cases, are the constraints holonomic or nonholonomic? Are the time dependent or independent?

Homework Equations

The Attempt at a Solution

In both cases the constraints are time dependent because they depend on the rotational velocity. In the first case, it is holonomic - you can define the bead's position in terms of time and the generalized coordinates.

However, I'm not sure about the second case. I think it might be holonomic as well, because we should be able to describe its constraints knowing only the hoop's equations of motion. The hoop has 2 degrees of freedom, theta and phi, and we define an origin we should be able to describe the particle's location at any time using just theta and phi.
Am I correct in my argument that the constraints in both cases are holonomic?

This question has my confidence completely stumped!

 

[haruspex]

we should be able to describe the particle's location at any time using just theta and phi.

Well, not the particle's exact location, but the constraints on it, yes. I agree they're both holonomic, though I'd never heard of the term until I saw this post :-).

 

[Mistro116]

If the bead were sliding with friction, would the constraint turn from holonomic to non-holonomic?

 

[haruspex]

If the bead were sliding with friction, would the constraint turn from holonomic to non-holonomic?

Not as I understand it. We are only concerned with the constraints on its location, right? I.e., where it physically could be. I don't see how friction changes that.
An example of a non-holonomic constraint would be an object moving on a curved surface, held down by gravity. Whether it stays on the surface would then depend on velocity.
I confess to being uneasy about this though. Even in the first case of the OP, whether it can reach the top of the hoop is limited by KE, so whether it's holonomic seems to depend on which constraints you choose to consider.

 

[andrien]

Holonomic constraints corresponds to case where eqn. are integrable.In case of friction,where the particle must also have to satisfy the eqn of surface,it turns out to be non holonomic.

 

[haruspex]

Holonomic constraints corresponds to case where eqn. are integrable.In case of friction,where the particle must also have to satisfy the eqn of surface,it turns out to be non holonomic.

Very surprised by that... what has integrability of the equation to do with the nature of the constraints? Is there a URL reference you can post?

 

[andrien]

Very surprised by that... what has integrability of the equation to do with the nature of the constraints? Is there a URL reference you can post?

There are a plenty of refrences.Why don't you start with wikipedia like here
http://en.wikipedia.org/wiki/Nonholonomic_system
If you want a book ,then 'Dynamics vol.2 by a.s. ramsey' or any other like of routh or whittaker has also given it in detail.

 

[haruspex]

There are a plenty of refrences.Why don't you start with wikipedia like here
http://en.wikipedia.org/wiki/Nonholonomic_system
If you want a book ,then 'Dynamics vol.2 by a.s. ramsey' or any other like of routh or whittaker has also given it in detail.

The reference I found was http://en.wikipedia.org/wiki/Holonomic_constraints, which gives a rather different impression.
The reference you give also results in a different answer for the second part of the OP: http://en.wikipedia.org/wiki/Nonholonomic_system#The_Foucault_Pendulum

 

[andrien]

The reference I found was http://en.wikipedia.org/wiki/Holonomic_constraints, which gives a rather different impression.
The reference you give also results in a different answer for the second part of the OP: http://en.wikipedia.org/wiki/Nonholonomic_system#The_Foucault_Pendulum

Your question was about the non-integrability relation to non-holonomic constraint which is clearly shown by the relation given on wiki.So far as the op question is concerned,I don't see any disagreement.I would like to know about that different impression.

 

[haruspex]

Your question was about the non-integrability relation to non-holonomic constraint which is clearly shown by the relation given on wiki.So far as the op question is concerned,I don't see any disagreement.I would like to know about that different impression.

The mention of integrability at the reference I found only says "if it's integrable then [constraints satisfying ... are holonomic]". It doesn't make integrability a requirement for holonomicity.
But maybe you can explain it to me this way: what constraint on the particle derives from the friction? The friction does not affect the set of places the bead can be at a point in time, and I don't know what other constraints one is supposed to consider.

Btw, back to the OP: the first case (wire rotating about an axis perpendicular to its plane) is clearly not time dependent.

 

[andrien]

The mention of integrability at the reference I found only says "if it's integrable then [constraints satisfying ... are holonomic]". It doesn't make integrability a requirement for holonomicity.
But maybe you can explain it to me this way: what constraint on the particle derives from the friction? The friction does not affect the set of places the bead can be at a point in time, and I don't know what other constraints one is supposed to consider.

Btw, back to the OP: the first case (wire rotating about an axis perpendicular to its plane) is clearly not time dependent.

Friction does not affect the situation here.But in a case,where a sphere rolls on a frictional surface.The point of contact must satisfy the zero velocity condition.With respect to which One can write the kinetic energy in terms of three euler angles θ,∅,ψ but the resulting eqn which one get by applying lagrange's eqn. will be incorrect because of non-holonomicity condition of eqn of rolling.In similar cases,one finds that when non-holonomic constraints are imposed,the number of variables used to describe eqn of motion are always greater than the numbers of degrees of freedom.Non-holonomic constraint also don't do work but they just involves a differential relation between velocities(generalised components) and generalised coordinates which can not be integrated because if they will ,then it will be possible to reduce the number of variables and system attains a holonomic state.
I do not want to continue this discussion further.

 

[haruspex]

Friction does not affect the situation here.But in a case,where a sphere rolls on a frictional surface.The point of contact must satisfy the zero velocity condition.

OK, so if it were static friction then it would be non-holonomic.
Thanks.