Virtual displacement vs. differential displacement

[MuIotaTau]

So what exactly is the difference between these two things? I understand their individual uses perfectly well, I think (or perhaps hope), but I'm a little unclear on what exactly a virtual displacement is. is it just a sort of imagined displacement that can't really occur, while a differential displacement is just an infinitesimally small but still physically significant displacement? Further, is there any sort of mathematical basis behind virtual displacement? Of course infinitesimal quantities can he defined using limits or hyperreal numbers; is there any analagous sort of construction for virtual quantites, or is it more of a physics-based idea? I'm sorry if I'm being unclear, it's possible I just don't know exactly what sort of question to ask.

 

[arildno]

Well, I am rather informal about this, but this is my take on it:
"Virtual" displacements is that class of legitimate displacements which is in accordance with the forces of constraint in the problem, but does NOT take into account the actual position, velocity and acceleration of the object in addition to the constraint forces.

The ACTUAL displacements are determined BOTH by constraint dynamics and the local kinematics of the object, whereas the VIRTUAL displacement is the wider class that merely look upon the constraint dynamics.

In short:
An actual displacement will necessarily also be (or have been) a legitimate virtual displacement in every case, but a virtual displacement might not be (/become) an actual displacement.

We can also say that with virtual displacements, we are concerned with the DEGREES OF FREEDOM of the system, rather than with the ACTUAL TRAJECTORY of the object.

 

[CAF123]

I am learning this terminology at the moment as well. The term 'virtual displacement' suggests to me something like an imaginary displacement but I don't think this is the case here. The definition I have is that a virtual displacement is a displacement in which constraint forces do no work consistent with the constraints of the problem.

If we take a rod revolving in a plane with constant angular velocity with a mass on the rod only allowed to move along the rod (this is the constraint), then the only virtual displacement (that is, a displacement consistent with the constraints) is for the particle to move along the rod. In polar coordinates, write $\vec{F}_{constr} = F_{constr} \underline{e}_{\phi}$ and so this is always perpendicular to the radial motion.

My notes then read, '...constraint forces may do work in a non-instantaneous change, including that of the actual motion, as the example above shows'
Can somebody explain what this means?

 

[arildno]

"The definition I have is that a virtual displacement is a displacement in which constraint forces do no work"
That constraint forces do no work is a SPECIAL case; and the one we wish to work with, primarily (that is how I read Goldstein, anyway).
First and foremost, constraint forces "kill off", in whatever manner, movements in certain directions.
Whether they do this in a manner that is consistent with energy conservation or not, is a SECONDARY (but very important!) consideration.

 

[MuIotaTau]

Okay, I think this is starting to make sense.

So if we take Caf's example, the only possible virtual displacement is motion along the rod due to the constraints of the system, but the actual, physical displacement may be, for instance, zero because the rod may be static, or it may be radially inward or outward depending on applicable forces. But we don't consider these factors initially because the virtual displacement is only an imagined, possible displacement whose only criterion is consistency with constraints. And so virtual displacements become physical displacements when you apply local kinematics.

Is that much right, at least? I'd like to make sure before I begin to mentally tackle the work aspect of it.

 

[arildno]

"But we don't consider these factors initially because the virtual displacement is only an imagined, possible displacement whose only criterion is consistency with constraints. And so virtual displacements become physical displacements when you apply local kinematics."

That is my take on it, yes.
The actual TRAJECTORY from a particular residing point will depend not only on the constraint forces and other forces active at that point, but also on the state of motion of the object. ACTUAL displacements are, of course, necessarily coupled to the trajectory.

Note that D'Alembert's principle precisely regards the system to be in a state of equilibrium, by regarding "-ma" as a generalized force, rather than as primarily a kinematic term. Thus, D'Alembert's principle, "couldn't care less" about motions (even though, obviously, it does in a brilliant manner)

In addition, you must achieve CLOSURE of your system of equations, in that the expression for the forces you retain have an explicit expression, for example in how they exactly depend upon the object's position (for example, in the form of a potential, like gravity or the Hooke's law for a spring).

 

[arildno]

Note that for holonomic constraints, where the allowable positions are declared through equations dependent on the object position, is the "principal effect" of the constraint forces, from which we may deduce the actual degrees of freedom, and thus the type of allowable virtual displacements the system has.

When we set up the principle for virtual work, we typically place the ADDITIONAL criterion on our system, namely that ALONG those legitimate displacement, the constraint forces do no work.

 

[MuIotaTau]

"But we don't consider these factors initially because the virtual displacement is only an imagined, possible displacement whose only criterion is consistency with constraints. And so virtual displacements become physical displacements when you apply local kinematics."

That is my take on it, yes.
The actual TRAJECTORY from a particular residing point will depend not only on the constraint forces and other forces active at that point, but also on the state of motion of the object. ACTUAL displacements are, of course, necessarily coupled to the trajectory.

Your take certainly makes an awful lot of sense, so I would definitely trust it!

Note that D'Alembert's principle precisely regards the system to be in a state of equilibrium, by regarding "-ma" as a generalized force, rather than as primarily a kinematic term. Thus, D'Alembert's principle, "couldn't care less" about motions (even though, obviously, it does in a brilliant manner)

So in some sense, D'Alembert's principle only concerns itself with virtual displacements by very cleverly "sweeping under the rug" (while still keeping track of) kinematics using an inertial force? In other words, all the kinematics get wrapped up into the inertial force and its then just treated like any other generalized force?

Ohhh, and so this is just extending the idea that virtual work is zero to dynamical systems by introducing the inertial force so that we may, in some sense, treat the system as static, correct?

In addition, you must achieve CLOSURE of your system of equations, in that the expression for the forces you retain have an explicit expression, for example in how they exactly depend upon the object's position (for example, in the form of a potential, like gravity or the Hooke's law for a spring).

Why exactly is that? It makes sense, of course, but I'm just curious as to the exact reasoning behind that, I guess.

Thank you both so much, by the way, this has been very enlightening overall!

 

[arildno]

1. "D'Alembert's principle only concerns itself with virtual displacements by very cleverly "sweeping under the rug" (while still keeping track of) kinematics using an inertial force? In other words, all the kinematics get wrapped up into the inertial force and its then just treated like any other generalized force?"
Yup!

2."Why exactly is that? It makes sense, of course, but I'm just curious as to the exact reasoning behind that, I guess."
I apologize for that point:
Whether you work with Newton or The Lagrangian formulation, you must have an equal number of equations as you have unkowns. That's maths, rather than physics, so sorry about that!

What I should have highlighted instead is that in the virtual work principle, you chip away unknowns (i.e, the constraint forces themselves, when they don't do any virtual work) that in Newton's formulation you would have include in a therefore larger system of equations.

The Lagrangian formulation exhibits the hidden core within Newton, by removing redundant (and complicating) elements.

 

[MuIotaTau]

1. "D'Alembert's principle only concerns itself with virtual displacements by very cleverly "sweeping under the rug" (while still keeping track of) kinematics using an inertial force? In other words, all the kinematics get wrapped up into the inertial force and its then just treated like any other generalized force?"
Yup!

2."Why exactly is that? It makes sense, of course, but I'm just curious as to the exact reasoning behind that, I guess."
I apologize for that point:
Whether you work with Newton or The Lagrangian formulation, you must have an equal number of equations as you have unkowns. That's maths, rather than physics, so sorry about that!

What I should have highlighted instead is that in the virtual work principle, you chip away unknowns (i.e, the constraint forces themselves, when they don't do any virtual work) that in Newton's formulation you would have include in a therefore larger system of equations.

The Lagrangian formulation exhibits the hidden core within Newton, by removing redundant (and complicating) elements.

No need to apologize at all, that all makes complete and total sense now! I've been trying to understand the basis of alternative formulations for some time now, and things are finally coming together, so I really appreciate that!