# The role of the virtual work principle

Lanczos' masterpiece "The Variational Principle of Mechanics" has, on page 76, the following statement:

Postulate A (virtual work): The virtual work of the forces of reaction is always zero for any virtual displacement which is in harmony with the given kinematic constraints.

This postulate is not restricted to the realm of statics. It applies equally to dynamics, when the principle of virtual work is suitably generalized by means of d'Alembert's principle. Since all the fundamental variational principles of mechanics, the principles of Euler, Lagrange, Jacobi, Hamilton, are but alternative formulations of d'Alembert's principle, Postulate A is actually the only postulate of analytical mechanics, and is thus of fundamental importance$^1$.

$^1$Those scientists who claim that analytical mechanics is nothing but a mathematically different formulation of the laws of Newton must assume that Postulate A is deducible from the Newtonian laws of motion. The author is unable to see how this can be done. Certainly the third law of motion, "action equals reaction", is not wide enough to replace Postulate A.

By "in harmony" he means forces that keep rigid bodies rigid, that is, that don't break the stuff you're studying. In the next chapter he proceeds to prove all of mechanics is deducible from Newton's Second Law and d'Alembert's principle, which is philosophically elaborate, but mathematically resumes to transforming $F = ma$ into $F-ma=0$, which turns dynamics into statics.

I have a feeling something is strange, here. Is the author stating that all analytical mechanics can be obtained from Newton's Second Law + Postulate A?

• Does this website help? Aug 1, 2016 at 20:48
• It does help clarifying why the Third Law and the Postulate are related, but I'm still looking for further clarification on the importance of the virtual work principle. Where does the First Law fit? Aug 1, 2016 at 20:55
• "The author is unable to see" is a very suspicious phrase. If the author has this conviction and if he has written "a masterpiece", then that masterpiece has to contain a mathematical proof that the statements are not equivalent. There is no philosophy in this, at all. If you want to see a well written book about mechanics that avoids all the philosophy, read Landau-Lifshitz Volume 1. Having said that, I think they pretty much bypass the general issue and focus on scenarios that are more important than the axiomatics of classical mechanics. Aug 1, 2016 at 21:32
• Indeed, "the author is unable to see" is extremely problematic. I agree and that's why I posted the question. But Lanczos' book covers ten times what Landau's cover, and is the total opposite of that you've suggested: its aim is to be philosophically oriented, and I'm curious about the axioms. Lanczos was not interested in how to calculate stuff, especially in a time where philosophy and physics were still quite mingled, but in finding out the "why"s in a time symplectic geometry didn't exist. Aug 1, 2016 at 21:40
• I am not saying that Lanczos is wrong, but one can't just wipe the issue off the table with such a remark. Technically it boils down to work being a matter of energy conservation, while the third law is about momentum conservation (the second law is silent on both issues). If you take Noether seriously, then Newton's laws are, indeed, incomplete. There should be an energy conservation law for closed systems. That there isn't has a good reason, of course: Newton includes thermodynamics, whereas rigid constraints do not. Aug 1, 2016 at 21:55

I’ll have a go at this, since we use it in pulley systems. Basically we consider an imaginary (infinitesimal) displacement dx, and then calculate the total ‘work’ done due to the forces acting on the system. The virtual work (W) theorem allows us to equate this W = 0.

This allows us to add 1 more equation to solve the question, which at times may be necessary or a faster way to solve problems. It is called ‘virtual’ work as there is no real work done, we are assuming one and equating it to 0 to solve a question.

You may want to refer to https://www.iitg.ac.in/kd/Lecture%20Notes/ME101-Lecture19-KD.pdf

I agree with the last statement made by the author indeed all mechanics can be derived from NLM, for example rotational mechanics, but we have devised other methods to make our work easier. ;)

You may want to have a look at a problem I just made to demonstrate how virtual work method can be used.

Hope it helps.

• On this site, please avoid screenshots. Your equations are hard to read and you should use the formatting tools made available here. Thanks. May 27, 2022 at 6:58
• @Miyase sorry, I didn’t know that. I will try to format next time. May 27, 2022 at 7:17
• Thanks for the effort! Unfortunately the original question has very little to do with solving problems in mechanics. I asked this question when I was still a first year PhD student, but even after 7 years I still marvel at the depth of the virtual work principle. It might well be the most general statement in the whole of physics! May 27, 2022 at 23:25
• @QuantumBrick guess I again tried to answer something beyond my level, I just wanted to give it a try to better my knowledge as well. May 28, 2022 at 13:33