What is a nonholonomic constraint?

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In summary, nonholonomic constraints are constraints in Mechanics that cannot be expressed as a functional relationship between coordinates. They often involve velocities and require the use of redundant coordinates and a Lagrange multiplier. A conservation law is not considered a nonholonomic constraint, as it is a result of the system's dynamics rather than a constraint imposed on it. Nonholonomic constraints can have a fundamental effect on a system's dynamics and stability.
  • #1
maburne2
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Hey guys,
What exactly does a nonholonomic constraint tell about a system. For instance I am working on a goldstein problem and it has raised the importance of interpreting what a constraint really does. I understand what a holonomic constraint is and what it tells me-for one the motion is bound-but I really do not know how to interpret the nonholonomic constraints. Are they criteria that specify how the system behaves, or what causes the system to have unbound motion? Any help would be greatly appreciated.

Thanks
 
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  • #2
maburne2 said:
What exactly does a nonholonomic constraint tell about a system. For instance I am working on a goldstein problem and it has raised the importance of interpreting what a constraint really does. I understand what a holonomic constraint is and what it tells me-for one the motion is bound-but I really do not know how to interpret the nonholonomic constraints. Are they criteria that specify how the system behaves, or what causes the system to have unbound motion? Any help would be greatly appreciated.
http://en.wikipedia.org/wiki/Nonholonomic
 
  • #3
Nonholonomic. Don't you just love these terms they use in Mechanics? How about scleronomous and rheonomous. Or the polhode. Whose only role in life it seems to roll without slipping on the herpolhode. I used to wonder: what horrible thing would happen if the polhode ever slipped? Nonholonomic constraints are what make Lagrangian Mechanics worth doing: they're hard to understand, but make it possible to solve difficult and interesting problems.

First, a holonomic constraint is one that can be expressed as a functional relationship between the coordinates: f(q1, q2,... ) = 0. The nice thing about such a constraint is that by simple substitution you can use it to eliminate one of the coordinates. However you're not required to: you can work a Mechanics problem using more coordinates than are strictly needed. For example take a point mass moving freely in a circle. You can work it the easy way using one coordinate Θ, or you can work it the hard way using two coordinates x and y, with the constraint that x2 + y2 = constant. Such a constraint is handled by introducing a Lagrange multiplier λ. And even that step is counterintuitive because now instead of solving a system with one variable, or even two variables you must solve a system with three: x, y and λ.

Well, a nonholonomic constraint is the other case: one that cannot be expressed as a functional relationship between the coordinates. Usually the velocities are involved. If you've understood the above paragraph, the only difference is that now you have no choice: you are forced to use redundant coordinates and introduce the Lagrange multiplier.
 
  • #4
Bill_K said:
Well, a nonholonomic constraint is the other case: one that cannot be expressed as a functional relationship between the coordinates. Usually the velocities are involved. If you've understood the above paragraph, the only difference is that now you have no choice: you are forced to use redundant coordinates and introduce the Lagrange multiplier.

Would a conservation law be regarded as a nonholonomic constraint? For example, take the simple harmonic oscillator: [tex]\mathcal L= .5(\dot{x}^2-x^2)[/tex] which would have the following relationship between velocities and coordinates: [tex]E=\mbox{constant}=.5(\dot{x}^2+x^2) [/tex]. It doesn't seem you can solve the latter equation for the position or velocity and substitute it back into the former equation and use Lagrange's equation. So would one have to use Lagrange multipliers with E as the constraint equation?
 
  • #5
Instead of being fully fixed or constrainted relative to something (holonomic), this variable can move, but only in a specific way. For example, ice skates can move relative to ice, but only in the direction of the blades (unless shredding occurs).
 
  • #6
A conservation law wouldn't be regarded as a constraint at all, because it's really a consequence of the dynamics. You don't need to enforce conservation of energy, it just falls right out of the dynamics as long as the Lagrangian doesn't have any explicit time dependence.
 
  • #7
Amen brotha. But your non-holonomic constraint may have a fundamental effect on system dynamics and stability.
 

1. What is a nonholonomic constraint?

A nonholonomic constraint is a type of constraint that restricts the motion of a system or object in a non-integrable way. This means that the constraint cannot be expressed as a simple mathematical relation between the position, velocity, and time of the system.

2. How is a nonholonomic constraint different from a holonomic constraint?

A holonomic constraint is a type of constraint that can be expressed as a simple mathematical relation between the position, velocity, and time of a system. In contrast, a nonholonomic constraint cannot be expressed in this way, making it more complex to analyze and model.

3. What are some examples of nonholonomic constraints?

Some common examples of nonholonomic constraints include rolling without slipping, skidding without sliding, and the motion of a pendulum with a fixed pivot point. These constraints can arise in various physical systems, such as vehicles, robots, and mechanical systems.

4. How do nonholonomic constraints affect the motion of a system?

Nonholonomic constraints impose restrictions on the possible motions of a system, limiting its degrees of freedom. This can result in non-intuitive behaviors, such as non-reversibility and non-conservation of energy, as well as challenges in control and optimization of the system's motion.

5. What are some applications of nonholonomic constraints in science?

Nonholonomic constraints have numerous applications in science and engineering, particularly in fields such as mechanics, robotics, and control systems. They are also important in understanding complex physical systems, such as the motion of celestial bodies and the behavior of fluids.

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