In Goldstein's Classical mechanics I found a proposition that I don't understand:

Similarly, the arbitrary virtual displacement $\delta \mathbf{r}_i$ can be connected with the virtual displacement $\delta q_i$ by $$\delta \mathbf{r}_i = \sum_j \frac{\partial \mathbf{r}_i}{\partial q_j} \delta q_j. \tag{1.47}$$ Note that no variation of time, $\delta t$, is involved here, since a virtual displacement by definition considers only displacements of the coordinates. (Only then is the virtual displacement perpendicular to the force of constraint if the constraint itself is changing in time).

I don't understand the part within parenthesis. Why is it important that there is no variation of time $\delta t=0$ in the definition of virtual displacement?

My understanding is that the force of constraint is perpendicular to the surface, while the virtual displacement must be tangent to it (and this should be the case also if the constraints change with time).

I'm not asking for the definition of virtual displacement. I already know that virtual displacement by definition occurs at $t=$ const so that the differential of $\mathbf{r}_i$ does not vary with time.


1 Answer 1



  1. Consider a point particle constrained to move on a rotating rod $$ {\bf r}~=~q \begin{pmatrix} \cos (\omega t) \cr \sin(\omega t) \cr 0 \end{pmatrix} .\tag{1}$$ There are 2 holonomic constraints, and 1 generalized coordinate $q$. Infinitesimal displacements are of the form $$ \delta {\bf r}~=~\delta q \begin{pmatrix} \cos (\omega t) \cr \sin(\omega t) \cr 0 \end{pmatrix}+ q\omega\delta t \begin{pmatrix} -\sin (\omega t) \cr \cos(\omega t) \cr 0 \end{pmatrix} .\tag{2}$$

  2. The constraint force belongs to the following 2-plane: $$ {\bf F}~\in {\rm span}_{\mathbb{R}}( \begin{pmatrix} -\sin (\omega t) \cr \cos(\omega t) \cr 0 \end{pmatrix},\begin{pmatrix} 0 \cr 0 \cr 1 \end{pmatrix} ).\tag{3}$$

  3. The infinitesimal displacements (2) are perpendicular to the constraint force (3) if we restrict to virtual displacements with $\delta t=0$.

  4. Concerning virtual displacements, see also e.g. this, this & this Phys.SE posts.


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