I?m having trouble understanding the difference between holonomic, scleronomic (fixed) and rheonomic constraints. Could someone give me some examples of these kinds of constraints?
Thanks
Let’s say you have a bunch of vectors to the things in your system, r[sub]1[/sub]-r[sub]n[/sub].
If you can describe the constraint as a function of all the vectors and the time, like
¦(r[sub]1[/sub], r[sub]2[/sub], r[sub]3[/sub], …, t) = 0
then the constraint is holonomic. If you can’t describe the constraint as such, it is nonholonomic.
An example of a holonomic constraint is the rigid body constraint - all parts of the system must remain fixed relative to one another. Another example would be a bead on a wire - it is constrained to move along the wire, so the equation describing the wire’s path also constrains the bead’s r vector.
An example of a non-holonomic constraint is a container constraining the motion of gas molecules, or the presence of a solid object in our space that the system cannot enter. For example, a solid sphere at the origin of radius a which our system cannot penetrate would give us the nonholonomic constraint
r[sup]2[/sup] - a[sup]2[/sup] >= 0
A more heinous example would be if the constraint is described by a differential equation - you may not be able to integrate the equation without solving your problem anyway.
Now, if the equation of constraint is explicitly time dependent, it is rheonomous. If it is not explicitly time dependent, it is scleronomous. Note that if time dependence is introduced as a side effect, it remains scleronomous - for example, if the motion of a bead on a wire causes the wire to move, it’s not explicit time dependence.
The holonomic/nonholonomic attribute is orthogonal to the rheonomic/scleronomic attribute.
Why does it matter? Holonomic constraints help you remove degrees of freedom from a system, and if you convert your coordinates appropriately you can have a much simpler system to solve. Nonholonomic constraints do not lend themselves to a systematic approach like this, so they suck.
I hope I didn’t just do your homework for you…
Reference: Classical Mechanics, Goldstein, page 13 or so.
Thanks douglips. I got most of that but I don’t quite understand what you mean by “explicitly time dependent”. Is that different than just having t as an idependent variable? I guess it is but could you give me an example of something that is “explicitly time dependent.”
“I hope I didn’t just do your homework for you…” Nope, I’m not even close to the end of the chapter. Also, I have the solutions manual so I never need help on how to do a problem I just sometimes need help on understanding stuff.
Explicitly time dependent means that the constraint has t as an independent variable.
Explicitly time dependent would be a bead moving on a wire, the wire is attached to a rotating turntable rotating at some velocity.
Non-explicitly time dependent would be a bead moving on a wire floating free in space. The wire may move in response to the bead’s motion, but does not have an explicit time dependence.
Non time dependent would be a bead moving on a wire that is always stationary.
Hope this helps…
Yes, it helps a lot. Thank you very much for your help. I’m going to print this out and put it in my Mechanics folder.