Is obtaining proper equation of motion from D'Alembert's principle a mere coincidence or there is some logic behind this?

This is asked because while we are finding the equations in a regime where the displacement is virtual, who guarantees that the obtained equations will do when the displacements are real, too?

  1. D'Alembert's principle $$ \sum_i (\mathbf{F}^{(a)}_i - \dot{\mathbf{p}}_i) \cdot \delta \mathbf{r}_i~=~0 \tag{1}$$ can be derived from Newton's laws and leads to Lagrange equations, see e.g. Goldstein, Classical Mechanics, Chapter 1.

  2. Superscript $(a)$ in eq. (1) stands for applied forces, cf. e.g. this Phys.SE post.

  3. Because the non-applied forces do no virtual work, we can rewrite (1) as $$ \sum_i (\mathbf{F}^{(\rm tot)}_i - \dot{\mathbf{p}}_i) \cdot \delta \mathbf{r}_i~=~0. \tag{2}$$ On one hand, eq. (2) is clear consistent with Newton's 2nd law $$ \dot{\mathbf{p}}_i ~=~ \mathbf{F}^{(\rm tot)}_i. \tag{3}$$ On the other hand, since virtual displacements probe in all allowed directions (which don't violate the constraints), Newton's 2nd law (3) follows from eq. (2).


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