1.  The mathematics of finding functions that extremize/optimize a given functional (variational derivative)
2.  The mathematics of  extremizing a functional subject to given constraints (isoperimetric problem)
3.  The mathematics of  extremizing a given functional at a boundary (end point problem)
4.  The mathematics of finding the shortest (or longest) path (geodesic problem)
5.  The mathematics of setting up the equations of mechanics from a single scalar (Hamilton's principle)
6.  The mathematics of phase fronts and their interference (Hamilton-Jacobi theory)
7.  The mathematics of tensors (multi-linear algebra)
8.  The mathematics of parallel transport (covariant differentiation)
9.  The mathematical classification of of parallel transport (torsion and curvature)
10.The mathematics of a strained elastic medium (metric-induced geometry)