1. The proof is based on the fact that, due to the conservation of energy, a moving point cannot move further from y than cN-½, which approaches zero as N → ∞.
2. By differentiable here we mean r times continuously differentiable; the exact value of r (1 ≤ r ≤ ∞)is immaterial (we may take r = ∞, for example).
3. A manifold is connected if it cannot be divided into two disjoint open subsets.
4. The authors of several textbooks mistakenly assert that the converse is also true, i.e., that if hs takes solutions to solutions, then h
s preserves L.
5. Strictly speaking, in order to define a variation δφ, one must define on the set of curves near x on M the structure of a region in a vector space. This can be done using coordinates on M; however, the property of being a conditional extremal does not depend on the choice of a coordinate system.