Understanding Non-Holonomic Constraints for Particle Motion

[Reshma]

A particle moves in the x-y plane under the constraint that its velocity is always directed towards a point on the x-axis whose absicissa is some given function of time f(t). Show that for f(t) differentiable, but otherwise arbitrary, the constraint is non-holonomic.

All I could infer from the above question is:
x = Cf(t)
C is a constant.
If the velocity is directed towards a point on the x-axis, is the same point?

Could someone guide me in the right direction?

 

[OlderDan]

A particle moves in the x-y plane under the constraint that its velocity is always directed towards a point on the x-axis whose absicissa is some given function of time f(t). Show that for f(t) differentiable, but otherwise arbitrary, the constraint is non-holonomic.

All I could infer from the above question is:
x = Cf(t)
C is a constant.
If the velocity is directed towards a point on the x-axis, is the same point?

Could someone guide me in the right direction?

I interpret the problem to mean that x = f(t). I see no need for the constant, since the only requirement is that f(x) be differentiable. And yes, no matter where the particle might be at some time, its velocity is always pointing toward that point x = f(t).

I have quite forgotten what makes a constraint holonomic, but I can imagine a bead sliding on a straight wire with one end attached at x=f(t) but free to rotate. The bead would be constrained to be moving only toward or away from the point x at any instant.

 

[Reshma]

I interpret the problem to mean that x = f(t). I see no need for the constant, since the only requirement is that f(x) be differentiable. And yes, no matter where the particle might be at some time, its velocity is always pointing toward that point x = f(t).

I have quite forgotten what makes a constraint holonomic, but I can imagine a bead sliding on a straight wire with one end attached at x=f(t) but free to rotate. The bead would be constrained to be moving only toward or away from the point x at any instant.

I'm strictly going by my textbook definition:
Holonomic constraints will have integrable terms, non-holonomic constraints will have non-integrable terms. So I think here I have to formulate a constraint equation which can be shown to be non-integrable and prove that the constraint is non-holonomic. Am I going right?

 

[Reshma]

I have quite forgotten what makes a constraint holonomic, but I can imagine a bead sliding on a straight wire with one end attached at x=f(t) but free to rotate. The bead would be constrained to be moving only toward or away from the point x at any instant.

Yes, this is a classic example of holonomic constraint. The constraint can be expressed in terms of its independent coordinates in the form:
f (r1, r2,...,t) = 0
But that can be done for a non-holonomic one.

 

[OlderDan]

Yes, this is a classic example of holonomic constraint. The constraint can be expressed in terms of its independent coordinates in the form:
f (r1, r2,...,t) = 0
But that can be done for a non-holonomic one.

I don't know what you are required to do to prove the constraint is non-holonomic, but I read a bit and reminded myself that holonomic constraints reduce the number of degrees of freedom of a system. This velocity constraint clearly does not do that. As a general rule, velocity constraints do not constrain the coordinates, so they are non-holonomic.

 

[thelaststand]

What are the other non-holonomic constraints? Correct me if I'm wrong but there's the leonomic and scleronomic right? What's the difference between the two?