I have started a book on theoretical mechanics which in the beginning is stated that in order to understand properly the material one should know manifold theory and differential geometry very well and there are other recommended books and also short appendix on this topic. It all was good until few pages further the motions of infinitesimal distance and displacements and virtual displacements started to occur. There was no single line about them in the appendix and also cannot find exact definition of these term in the given books. So my question is how these notions are well mathematically defined in terms of differential geometric structures (maps from where to where and why and how) and where I can find further information about them? EDIT: If there is way not to use infinitesimals but standard notions, it would be better.
If what you are primarily interested in is mathematical physics then it may make sense to work with actual infinitesimal displacements that can be developed as a rigorous basis for differential geometry, as in this publication. A more detailed version of this approach can be found in this set of lecture notes currently being used in a course on true infinitesimal differential geometry.