D'Alembert's Principle: rigorous formulation using notions from modern differential geometry

Question

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.

Answer 1

More a long comment than an answer, though I hope to give some useful references. Maybe the question is a little ambiguous, since contemporary expositions of rational (meaning "build in an axiomatic way") mechanics have almost always a strong geometric flavor. The book of V. I. Arnol'd is a standard example of this trend, and while his exposition of the D'Alembert-Lagrange principle ([1] §21, pp. 91-97) is not strictly written in the language of differential geometry, he nevertheless uses systematically varieties and the tangent manifold concept. Perhaps a more differential geometric exposition is the one presented by A. Prastaro in his book [4] (§3.4, pp. 364-369): apart from the use of the concepts of tangent space and manifold he defines virtual works and constraints as differential $1$-forms. However, while closer to Arnol'd's approach the to Pràstaro's one, my advice is to have a look ant the beautiful and important paper [2] of Carbonaro and Starita (and its errata [3] consisting in a missing page from [2]): it starts with a comprehensive historical survey, then it goes on by offering a formulation and a proof of the under the hypothesis that

1. the constraints are holonomic and time independent and
2. the forces of constraint satisfy an ideality condition.

Thus, while perhaps not having a sufficient differential geometric flavor, [2] can help by guiding you towards a definition well suited for your research need. Well, my two cents.

References

[1] Vladimir I. Arnold, Mathematical methods of classical mechanics, second edition, translated by K. Vogtmann and A. Weinstein. (English) Graduate Texts in Mathematics. 60. New York-Heidelberg-Berlin: Springer-Verlag. xvi+516, MR0997295.

[2] Bruno Carbonaro, Giulio Starita, "The principle of virtual velocities" (English), in Remigio Russo (ed.), Classical problems in mechanics, Rome: Aracne Editrice. Quaderni di Matematica 1, 1-95 (1997), ISBN 88-7999-187-6, MR1636284, Zbl 0992.70003.

[3] Bruno Carbonaro, Giulio Starita, "An integration to ”The principle of virtual velocities”* (English), in Vsevolod A. Solonnikov (ed.), Recent developments in partial differential equations. Rome: Aracne Editrice Quaderni di Matematica 2, 255-258 (1998), ISBN 88-7999-210-4, Zbl 0992.70004.

[4] Agostino Prástaro, Geometry of PDEs and mechanics (English), Singapore: World Scientific, pp. x+750 (1996), ISBN: 981-02-2520-2, MR1412798, Zbl 0866.35007.