# D'Alembert's principle

Actually I have some troubles to understand what this principle is all about, so I want to use the simple pendulum in order to get the idea. Since I have read a few passages that dealt with this concept, I would ask anybody who wants to help me, that he tries to answer my questions instead of rather talking about this principle in general.

So imagine a simple pendulum, then we have $F = - mg e_z +F_{\text{tension}}$. Actually, $-mge_z$ is our applied force and $F_{\text{tension}}$ our constraint force, right?

Now we define our virtual displacements in such a way that they are: infinitesimally small, we freeze the dynamics of the system (so no time passes by) and the displacement is fine with the constraint forces.

I am thinking about this in the following way: We want to have a tool at hand, that tells us, how our particle can move in the current configuration of forces, which is why we don't want any time to elapse, that would change the forces velocities and so on. Further, only tiny small displacements are more or less exactly caused by the current configuration of the constraint force and the applied force, which is why we want to look at the motion that is in a small neighbourhood happening.

That is should be reconcilable with the constraint force is probably straightforward, if we assume to describe physics with this concept.

Now comes my problem:

We are looking at a system in equilibrium $$W = \sum_i F_i \cdot \delta x_i =0,$$ so a system that does not get or loose energy. Then we have: $$W = \sum_i F_{\text{applied},i} +F_{\text{constraint},i} \cdot \delta x_i =0,$$ Assuming $$W = \sum_i F_{\text{constraint},i} \cdot \delta x_i =0,$$ then we get $$W = \sum_i F_{\text{applied},i} \cdot \delta x_i =0.$$ Now am I right, that this is wrong for the simple pendulum (because there the dot product of gravitational force and motion is not zero) but only right for systems that do not move like a book on a desk? (that would explain to me why it is supposed to apply for static systems.)

Then D'Alembert's principle is completely strange to me: It says $$\sum_{i} (F_{\text{applied}} - F_{\text{total}} )\cdot \delta x_i = 0$$ I mean, if we insert $$F_{\text{total}} = F_{\text{applied}} + F_{\text{constraint}}$$ we arrive just at $$\sum_{i} F_{\text{constraint}} \delta x_i = 0$$ which is what we already knew and since apparently both expressions are equivalent, so D'Alembert's principle just says: constraint forces do not produce any physical work? As this was our input, this sounds like: I prove my premise and therefore I must be wrong somewhere.

The problem is that you assume the system is in equilibrium in your first line. Apparently the pendulum is not in equilibrium if the dot product of gravity and motion is not zero.

But actually d'alembert principle states the following for general cases,
$$\sideset{}{}\sum_{i}(F_i-\dot{p})\cdot \delta x_i=0$$ So we have $$\sideset{}{}\sum_{i}(F_{applied,i}+F_{constraint,i}-\dot{p})\cdot \delta x_i=0$$ And we only consider the cases in which the contranint forces do not do work. So $$\sideset{}{}\sum_{i}F_{constraint,i}\cdot \delta x_i=0$$ And thus we have $$\sideset{}{}\sum_{i}(F_{applied,i}-\dot{p})\cdot \delta x_i=0$$

And this is the d'alembert principle.