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In summary: 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'"This is true, but there are other criteria as well- for instance, the constraint forces must be consistent with energy conservation.

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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.

"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.

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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?

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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.

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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.

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"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).

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).

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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.

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arildno said:"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!

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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.

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.

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arildno said:

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!

Virtual displacement refers to a hypothetical displacement that is used to analyze the behavior of a system, while differential displacement refers to an actual small change in position of a body in a system.

Virtual displacement is often used to calculate differential displacement in order to analyze the system's behavior and predict its future state.

No, virtual displacement and differential displacement are two distinct concepts and cannot be used interchangeably. They serve different purposes in analyzing the behavior of a system.

Virtual displacement is commonly used in theoretical physics and engineering to analyze the motion of particles and structures, while differential displacement is used in fields such as mechanics and fluid dynamics to study the behavior of physical systems.

In mechanics, virtual displacement is used to analyze the equilibrium and stability of a system by considering a small, hypothetical displacement of the system and observing its response. This helps in predicting the behavior of the system under different conditions.

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