# D'Alembers Principle - further explanation [duplicate]

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?

• „ 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
– Eli
Mar 27, 2020 at 15:06
• So in your example you don’t have constraint force at all
– Eli
Mar 27, 2020 at 15:13
• @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 ? Mar 27, 2020 at 15:21
• equation (2) is a dot product between the constraint force and the virtuelle displacement which is not a function of time
– Eli
Mar 27, 2020 at 15:30
• @Eli If (2) doesn't hold, what would the revised equation be for a differential increase in time ? Mar 27, 2020 at 15:34