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In question : Why is the d'Alembert's Principle formulated in terms of virtual displacements rather than real displacements in time? there is a response :

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Perhaps a simple example is called for.

Example. In 2D consider a point mass $m$ with position ${\bf r}=(x,y)$ that is constrained to move on a frictionless vertical rod, which in turn has pre-determined horizontal position $$x=f(t),\tag{1}$$ where $f$ is a given function of time $t$. In other words, eq. (1) is a holonomic constraint, and there is one generalized position coordinate $q\equiv y$, i.e. one degree of freedom. The constraint force ${\bf F}^{(c)}$ is horizontal. The virtual displacements $\delta q \equiv\delta y$ are by definition vertical with $\delta t=0$, leading to that the constraint force ${\bf F}^{(c)}$ does no virtual work$^1$ $$ {\bf F}^{(c)} \cdot \delta {\bf r}~=~0.\tag{2} $$ However if we allow $\delta t\neq 0$, then eq. (2) will no longer hold, and this implies that we can no longer derive d'Alembert's principle from Newton's 2nd law.

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$^1$It is tempting to call eq. (2) the Principle of virtual work, but strictly speaking, the principle of virtual work is just d'Alembert's principle for a static system. For d'Alembert's principle, see also this and this related Phys.SE posts.

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$\boldsymbol {My}$ $\boldsymbol {question}$ is if $\delta t\neq 0$ why will eq. (2) no longer hold?

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  • $\begingroup$ „ eq. (1) is a holonomic constraint“ I don’t think so , holonomic constraint is something like this $x^2+y^2=L^2$ for example $\endgroup$
    – Eli
    Mar 27, 2020 at 15:06
  • $\begingroup$ So in your example you don’t have constraint force at all $\endgroup$
    – Eli
    Mar 27, 2020 at 15:13
  • $\begingroup$ @Eli - the response I quoted has a constraint force, so are you suggesting that the response I quoted is incorrect ? In any case what is the exact difference between the virtual time =0 and an actual displacement in time behaviour in this case ? $\endgroup$ Mar 27, 2020 at 15:21
  • $\begingroup$ equation (2) is a dot product between the constraint force and the virtuelle displacement which is not a function of time $\endgroup$
    – Eli
    Mar 27, 2020 at 15:30
  • $\begingroup$ @Eli If (2) doesn't hold, what would the revised equation be for a differential increase in time ? $\endgroup$ Mar 27, 2020 at 15:34

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