Is there a rigorous definition of D'Alembert's principle of virtual dynamic work in the language of differential geometry? Some questions I'm hoping to answer are:

1. How to view the configuration space of a set of particles as a smooth manifold.
2. The definition a virtual displacement as a smooth manifold object such as a tangent vector or 1-form.
3. A rigorous statement of the principle of virtual work in the language of tangent vectors or 1-forms.
4. A rigorous derivation of Lagrange's equations from this principle.

For reference, a standard physic presentation/derivation is given in §6.3 of Cline - Variational principles in classical mechanics.