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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?

Qmechanic
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QuantumBrick
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2 Answers2

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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.

enter image description here Hope it helps.

PSR_123
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I assume the author is referring to constraint forces as the forces of reaction. For example, if a book is placed on a coffee table (XY plane), the reaction force of the table (along Z-direction) keeps the book on it. The force due to the constraint in this case depends on the weight of the book. In a general problem, it may be incredibly difficult calculate the force due to constraint.

Now coming to the question; the way I understand it is as follows:

Virtual displacement is any potential diplacement of the particle. In the above the above example, the particle can move on the coffee table (XY plane) and hence any vector on this plane is a virtual displacement. Thus the force of constraint (along Z-axis) is perpenticular to the virtual displacement. Thus, the virtual work which is the dot product between any vector on that plane and the force of constraint vanishes. This allows you to get rid the of the complex and often unknown constraint forces from equations.

It is not possible to get all mechanics from principle of virtual work. For example, you can treat the frictional forces as a constraint force. This force is parellel to any potential displacement of the particle and hence principle of virtual work do not apply.

Jo Carlo
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