D'alembert's Principle: Doubts Explained

[pccrp]

/I'm having some doubt with D'alembert's Principle. The principle states that \sum_{i}(\vec {F}_i - \dot{\vec{p}}_i)\delta\vec{r}_i=0 but does that mean that each term of the summation must vanish too, or just the sum does? I know that mathematically there's no need that each term shall vanish, but does physical considerations requires them to vanish separately?

 

[vanhees71]

The point of d'Alembert's principle is to use it for constraint systems. If the constraints are all holonomic, you can introduce independent coordinates q_k writing \vec{r}_i=\vec{r}_i(q_1,\ldots,q_f), where f is the number of independent degrees of freedom. Then you can vary the q_k independently from each other, and this gives you equations of motion in terms of these variables.

Another method is to keep the Cartesian coordinates \vec{r}_i and implement the constraints with help of Lagrange multipliers. Then you can vary the \delta \vec{r}_i independently. This leads to equations of motion, where additional forces from the constraints are taken into account.

 

[UltrafastPED]

Let's take this in two steps:

The Principle of Virtual Work tells us that constraints do no work.

D'Alembert's Principle permits the impressed forces of dynamical systems to be handled in the same way as constraints - the application of this is to find the unknown trajectories caused by the impressed forces via the calculus of variations - thanks to Euler and Lagrange.

For a system in equilibrium the independence of the coordinates tells us that each of the terms must be zero; for dynamics it is only the sum which must be zero.

 

[pccrp]

But why in dynamic situations only the sum must be zero, and not each member of the sum?

 

[UltrafastPED]

In the case of equilibrium each variation has only a single coefficient: the value for the force of that constraint. In linear algebra it is shown that if the variables (coordinates) are linearly independent then the sum of a set of coefficients times the independent variables can only sum to zero when all of the coefficients are identically zero.

In the dynamical case you have a difference which must be zero ... but now the expression depends upon the trajectories via Newton's second law of motion: so yes, each term must go to zero, and this is the variational principle which is used to find those trajectories.

Sorry if my earlier statement was misleading and unclear.