classical mechanics - Confusion with Virtual Displacement

I have just been introduced to the notion of virtual displacement and I am quite confused. My professor simply defined a virtual displacement as an infinitesimal displacement that occurs instantaneously in the configuration space, but this doesn't make any mathematical sense to me. Is there a way of defining the notion of virtual displacement, and by extension virtual work and generalized forces, in a way that is more mathematically rigorous? Assuming purely geometric holonomic constraints as well as purely conservative forces in the definition is acceptable.

As a side note, I have seen some sources give vague statements that the virtual displacements are elements of the tangent space of the manifold of constraint and other sources suggest that the virtual displacements are a result of the total derivative of the position of a particle. A rigorous definition of virtual displacement should make the connection between these two concepts more clear.

Answer

1. 
   

   Let there be given a manifold $3N$-dimensional position manifold $M$ with coordinates $({\bf r}_1, \ldots, {\bf r}_N)$. Let the time axis $\mathbb{R}$ have coordinate $t$.

   
   


2. 
   

   Let there be given $m\leq 3N$ holonomic constraint functions $$f^a: M\times \mathbb{R} ~\to~\mathbb{R}, \qquad a~\in~\{1,\ldots, m\}. $$ The constraint functions $(f^1,\ldots, f^m)$ are usually assumed to satisfy various regularity conditions, cf. my Phys.SE answer here. Moreover, they are assumed to be functionally independent, and the intersection of their zero-level-sets $$C~:=~\bigcap_{a=1}^m (f^a)^{-1}(\{0\})~\subseteq~ M\times \mathbb{R}$$ is assumed to form a submanifold of dimension $3N+1-m=n+1$, which we will call the constrained/physical submanifold. Let $(q^1, \ldots, q^n,t)$ be coordinates on $C$. The $q$'s are known as generalized coordinates.

   
   



3. 
   

   Given a point $p\in C$ with coordinates $(q^1_0, \ldots, q^n_0,t_0)$, then the $n$-dimensional submanifold of finite virtual displacements at $p$ is $$ V~:=~C \cap (M\times \{t_0\}). $$ So in a nutshell, a finite virtual displacement is a displacement of position that doesn't violate the constraints and is frozen in time. See also this related Phys.SE post.

   
   


4. 
   

   The heuristic notion of infinitesimal variations can in this context be replaced with tangent vectors. So the tangent space of infinitesimal virtual displacements at $p$ is $$ T_pV. $$ Concerning the use of infinitesimals in physics, see also this and this Phys.SE posts.