Prerequisites needed to grasp virtual displacement

[amjad-sh]

In most of classical mechanics textbooks,the concept of virtual displacement and virtual work is not represented deeply.They just focus in how the method should be used without explaining from where this principle came and why scientists created it in the first place.

So to grasp this concept do I need to have a solid background in differential geometry and topology?

I'm saying this because I read in Wikipedia that:" virtual displacement arises from differentiating with respect to the parameter ε enumerating paths of the motion varied in a manner consistent with the constraints (thus pointing at a fixed time in the direction tangent to the constraining manifold)".

The manifold is a topological concept and it is related also to differential geometry.

In fact I'm trying to grasp Lagrangian mechanics,and I think that virtual displacement and work are like a cornerstone to enter deeply in the subject.
Thanks in advance!

 

[gleem]

This is coming from an experimental physicist who has had little contact with mechanics for some time.

They just focus in how the method should be used without explaining from where this principle came and why scientists created it in the first place.

It wasn't created in so much as it was discovered. Virtual displacement is a conceptual idea. It is a displacement in which the time is held constant. since a real displacement would involve a small evolution in time. Galileo was the first to note that in a condition of static equilibrium if a small displacement of objects subject to constraints was made the work done in the system was zero. This was later recognized by J Bernoulli to be of universal applicability for all systems is static equilibrium. Thus ΣF_i⋅δx_i = 0 Where δx_i are the virtual displacements for the respective forces. the principle of virtual displacement is valid in that it gives the same result as a conventional method of resolution of forces does.

So to grasp this concept do I need to have a solid background in differential geometry and topology?

No..

As far as Lagrangian mechanics is concerned the use of virtual displacement maintains it initial meaning, a change in a coordinate without a concomitant change in time. Beyond that I see nothing to extend its applicability. Some texts do not use the term virtual displacement in their development of the Lagrangian as it would be redundant.
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[amjad-sh]

Thanks gleem for your explanation, but I have some questions.

Galileo was the first to note that in a condition of static equilibrium if a small displacement of objects subject to constraints was made the work done in the system was zero.

How Galileo discovered this, and what are the experiments made that revealed this fact?

This was later recognized by J Bernoulli to be of universal applicability for all systems is static equilibrium. Thus ΣFi⋅δxi = 0 Where δxi are the virtual displacements for the respective forces.

Did j Bernoulli conclude this mathematically?and what domain of math he used to recognize it? since virtual displacement is "virtual" and can't be formulated using experiments as we can't observe virtual things.

 

[gleem]

I don't know the details but presumably he recognized the principle by just considering an object on an inclined plane held in place by another object connected to it and hanging off the edge of the plane. Bernoulli recognized the universality of this idea no derivation just that it works. Think of a virtual displacement as a thought experiment.