# Virtual displacements

From Goldstein's Classical Mechanics,

A virtual displacement of a system refers to a change in the configuration of the system as the result of any arbitrary infinitesimal change of the coordinates $\delta r_i$, consistent with the forces and constraints imposed on the system at the given instant $t$.

From Calkin's Lagrangian and Hamiltonian Mechanics

Freeze the system at some instant of time $t$, then imagine the particles displaced amounts $\delta r_i$ consistent with the conditions of constraint. This is called the virtual displacement.

I have seen some of the posts regarding virtual displacements but I have a question that doesn't seem to be answered by other posts. Suppose we have an inclined plane and a particle is sliding on the frictionless surface of the plane, so there is a normal force pointing perpendicular from the surface (outward).

From the definition of virtual displacements, it stated that virtual displacements should be consistent with the constraint equation. What does this exactly mean? Does that mean the virtual displacement should always "move" through the constraining object e.g. surface, rod, etc. such that it is always perpendicular to the constraining force? So in this case the virtual displacement is along the surface so that the dot product of the normal force and virtual displacement is zero.

That means I should just think of virtual displacements as those imaginary displacements that "I should set" to be perpendicular to the constraint forces. Take note "I should set".

## 1 Answer

The main difference between "real" displacements and "virtual" displacements arise when the constraint itself is time-dependent.

For example, you have an incline that is accelerating (I am saying accelerating because a moving side can be cancelled with galilei transform, accelerating one cannot; but you should imagine simply a moving incline). Then because the slide will move, any instantenous (!!!!) real displacement $d\mathbf{r}$ will not be tangential to the incline. It will consist of a $\delta\mathbf{r}$ displacement that is tangential along the incline, and a displacement $\Delta\mathbf{r}$ caused by the motion of the incline, so $d\mathbf{r}=\delta\mathbf{r}+\Delta\mathbf{r}$.

The $\delta\mathbf{r}$ here is what is called a virtual displacement. Because it doesn't take into account the time-evolution of the constraint (the motion of the incline), it can be calculated by "freezing time" and in the frozen landspace, you evaluate the possible displacements of the system that are allowed by the contraints (eg. in this case, tangent to the incline), hence the Calkin definition.

• So basically, virtual displacements are displacements "always along" the constraining body (e.g. surface, shaft of a rod, etc) since that is the only way for it to be consistent with the constraint? – mathemania Sep 1 '17 at 9:37
• I am not completely sure as there might be pathological cases (nongeometrical contraints exist), but right now I cannot think of any counterexample. – Bence Racskó Sep 1 '17 at 9:44
• Would it be safe to say that it is for me to judge whether the principle of virtual work holds for a certain physical situation where what I'm saying above holds (virtual displacements are perpendicular to the forces of constraint)? Otherwise, the principle of virtual work does not hold. – mathemania Sep 1 '17 at 9:48