Talk:Virtual displacement

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[edit] Main idea

The main idea of virtual displacement is that it is arbitrary, or even imaginary. For practical application to actual physical system we usually consider only virtual displacements that meet the physical constraints; however, for generality, constraints should not be part of the definition of virtual displacements.

Hmm, well the way I see it, if virtual displacements are only ever really discussed in the context of constrained systems, then to say that they are really more general is merely a philosophy. I'd say that they're only imaginary in the sense that energy is imaginary or action is imaginary. Both are man-made constructs. Also, if virtual displacements really are completely arbitrary, what distinguishes them from plain ol' displacements? - Miles

But virtual displacements aren't only discussed in the context of constrained systems! For instance, there is a straightforward derivation of the conservation of energy using virtual displacements that doesn't require constraints. On the other hand, it is important to understand how virtual displacements behave in the presence of constraints. Trevorgoodchild 15:47, 19 September 2007 (UTC)

[edit] Virtual Displacements as a Special Case of Infinitesimal Displacements

I disagree with the current assertion that virtual displacements are a special case of infinitesimal displacements - it has always been my understanding that it is in fact the other way around. The issue is a little tricky to discuss, because there are two ways in which a displacement can be considered "arbitrary." Virtual displacements are not "arbitrary" in that they must satisfy the given constraints, but they are "arbitrary" in that they can happen in any direction that satisfies the given constraints. (Another way of thinking about it is that a virtual displacement occurs in every direction that satisfies these constraints.) On the other hand, "actual" displacements are not arbitrary in either sense: they must satisfy the given constraints and they occur in one particular direction since they depend completely on, e.g., an infinitesimal displacement in time.

Here's Lanczos' discussion of virtual vs. actual displacements from The Variational Principles of Mechanics (chapter 2, section 2):

"A "variation" means an infinitesimal change, in analogy with the d-process of ordinary calculus. However, contrary to the ordinary d-process, this infinitesimal change is not caused by the actual change of an independent variable, but is imposed by us on a set of variables as a kind of mathematical experiment. Let us consider for example a marble which is at rest at the lowest point of a bowl. The actual displacement of the marble is zero. It is our desire, however, to bring the marble to a neighboring position in order to see how the potential energy changes. A displacement of this nature is called a "virtual displacement." The term "virtual" indicates that the displacement was intentionally made in any kinematically admissible manner. ... It was Lagrange's ingenious idea to introduce a special symbol for the process of variation, in order to emphasize its virtual character. This symbol is $δ$ . The analogy to $d$ brings to mind that both symbols refer to infinitesimal changes. However, $d$ refers to an actual, $δ$ to a virtual change. ... Note the fundamental difference between $δ y$ and $d y$ . Both are infinitesimal changes of the function $y$ . However the $d y$ refers to the infinitesimal change of the given function $y = f (x)$ caused by the infinitesimal change $d x$ of the independent variable, while $δ y$ is an infinitesimal change of $y$ which produces a new function $y + δ y$ ."

In fact, Lanczos uses the fact that the actual displacements (the $d$ s) are a special case of the virtual displacements (the $δ$ s) when deriving the conservation of energy (chapter 4, section 3):

"...let us now dispose of the $δ R k$ - which mean arbitrary tentative variations of the radius vector $R k$ - in a special way. Let these tentative displacements coincide with the actual displacements as they occur during the time $d t$ . This means that we replace $δ R k$ by $d R k$ , which is merely a special application of the variation principle."