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 [here](https://phys.libretexts.org/Bookshelves/Classical_Mechanics/Variational_Principles_in_Classical_Mechanics_(Cline)/06%3A_Lagrangian_Dynamics/6.03%3A_Lagrange_Equations_from_dAlemberts_Principle).