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
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(q_{1}, q_{2},... ) = 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 x^{2} + y^{2} = 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.
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?
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).
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.
Amen brotha. But your non-holonomic constraint may have a fundamental effect on system dynamics and stability.