# Virtual differentials approach to Euler-Lagrange equation - necessary?

I'm currently teaching myself intermediate mechanics & am really struggling with the d'Alembert-based virtual differentials derivation for the Euler-Lagrange equation. The whole notion of, and justification for, using 'pretend' differentials over a time interval of zero just isn't sinking in with me. I also notice that not all textbook authors invoke it, so I'm wondering how necessary it is, given that Hamilton's principle gets us to E-L just fine. So, will I ever need this virtual displacement/work approach for something other than a route to E-L, or can I safely wave bye-bye to it?

-
I think I've seen it actually used in an engineering calculation someplace, but if you're interested in, say, high energy physics you can certainly drop it. – alexarvanitakis Feb 27 '13 at 1:29

I have never really needed to use the principle of virtual work in practice (either as a student, or in research), so I personally think it's safe to say that it's not necessary that you spend too much time on this.

Having said this, there is a mathematically rigorous formulation of virtual displacements and virtual work that is very much obscured (at least in my opinion) by most physicists. This formulation basically involves virtual displacements considered as tangent vectors to the configuration manifold of the system. There is a wonderful description of all of this in the book Physics for Mathematicians: Mechanics 1 by the author Spivak, but I think most of the content in that book regarding virtual work can be found in the set of notes Elementary Mechanics from a Mathematician's Viewpoint on which the book was based. See especially page 63 and the statement of d'Alembert's Principle.

-

For all practical applications in modern physics (with the possible exception of certain applications in mechanical engineering), one can safely forget about d'Alemberts principle and the principle of virtual work. One usually only needs the principle of stationary action, and the Euler-Lagrange equations.

However, if one belongs to the group of people, who finds Newton's laws more intuitive than Lagrange equations, then it is, for purely theoretical reasons, immensely satisfying to see Lagrange equations derived from Newton's laws. This is e.g. done in the first chapter of Herbert Goldstein, Classical Mechanics.

An important element in this derivation is to show that a large class of constraint forces do no virtual work, leading to D'Alembert's principle.

[Only the opposite (almost trivial) derivation from Lagrange equations to (an unconstrained) Newton's second law, is usually shown in elementary classes of classical mechanics.]

Finally let us mention, that virtual paths play a profound role in the path integral formulation of quantum mechanics.

-

It is not the question of bye-bye or Hello-Hello :) This principle shows why E-L equations are useful -- because you CAN not bother with constraint forces, and write all funny problems of mechanics directly using generalized forces.

-