# A Principle of Virtual Work

#### Trying2Learn

Summary
Virtual Work, Calculus of Variations, Hamilton's Principle
Hello

May I ask: "What is the guiding philosophy of the Principle of Virtual Work?"

I do understand it an how to use it, in classical mechanics.

(And I will openly admit that I do not entirely understand what I am looking for, or what I am asking.)

But what is happening at this moment in history? What is the motivating factor for this development?

I understand the Calculus of Variations and Hamilton's Principle. I understand how the Principle of Virtual Work is an offshoot.

But this entire Principle of Virtual work (before it was endowed with mathematical rigor) seems so unusual. What even motivated people to come up with this in the first place?

"account for the forces and the virtual displacement for boundaries that are not fixed." Yes, I get it, but it just seems like the most unexpected thing to come up with.

(I wish I could explain exactly what I am looking for.... something at the intersection of philosophy, intuition, sociology, motivation, math...)

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#### vanhees71

Gold Member
I think, it's pretty intuitive to think about it as a very natural thing to do: The main reason for d'Alembert's principle beyond "naive" Newtonian mechanics is to handle systems with constraints, where pure geometrical, holonomic constraints are the most simple case.

As an example suppose, you've mounted a wheel on your bicycle. Now, what are doing to check, whether it's correctly fixed and that it's freely rotating as it shoud? Very simple: You check it out by excerting some appropriate forces at it to see how it reacts, and you get a pretty good intuitive feeling from these reactions, whether all is fine. That's precisely the virtual displacements you also apply in your formulae.

#### Trying2Learn

thanks you. That gave me more insight. And thank for the extra regarding the constraints.

#### sophiecentaur

Gold Member
Summary: Virtual Work, Calculus of Variations, Hamilton's Principle

(I wish I could explain exactly what I am looking for.
This may be worth looking at. Sorry if it's way too simplistic.
I haven't used Virtual Work in anger since A level. That was in the context of networks of struts and wires (or any other structure in equilibrium). As I remember, it was an alternative to using Forces and, instead, considered work and energy for small displacements. When a system is equilibrium, any displacement involves positive work done on it. I seem to remember writing down a differential equation and solving for a minimum. That would give values for stress in the structure without using vectors.
Despite never having actually used that approach formally, I took a lesson from the advantage, at times, of using Energy and not Forces to solve problems. Hence Hamiltonians come into play.

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#### vanhees71

Gold Member
Yeah, after you get used to the variational principles, especially Hamilton's least (or rather stationary) action principle, you'll ask why the heck they have tormented you with the much more tedious force concept in the beginning.

I remember once in the introductory physics lab we had the Atood machine as an experiment. They asked us to prepare the theory beforehand. What I did was simply to write down the treatment with d'Alembert's principle. The "colloquium" before the lab was the shortest the tutor ever experienced ;-)).

Also a highlight last semester was, when the students told me, why I hadn't explained Hamilton's principle earlier to them. They also thought that it's much simpler to calculate mechanics problems using appropriate generalized coordinates instead of complicated forces. Of course, whenever you need the reaction forces to constraints (which is of course very important to check in practice!), it's also much simpler to calculate them after you got the solution of the equations of motion than having to find them at the same time.

#### anorlunda

Mentor
Gold Member
Yeah, after you get used to the variational principles, especially Hamilton's least (or rather stationary) action principle, you'll ask why the heck they have tormented you with the much more tedious force concept in the beginning.
Yes indeed. It seems counter-intuitive that an advanced technique might be significantly easier than the elementary basic approach. The only analogy I can think of is that Laplace transforms are taught only at the end of differential equations courses.

#### sophiecentaur

Gold Member
It seems counter-intuitive that an advanced technique might be significantly easier
Except that everyone can feel a force and appreciate what it does. Change in potential, although it may be 'easy' to calculate, is a much more sophisticated concept. They have to start with forces in Mechanics. If they didn't, most of the lower school classes would never even get themselves started.
It's sometimes very easy to look back down the road and feel that the route there was obvious. Lets face it, PF is deluged with demands to explain Electric Potential in terms of Forces and that just goes to prove my point.

"Principle of Virtual Work"

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