The third chapter of Arnold's Mathematical Methods of Classical Mechanics deals with variations (relevant passages included at the end), yet no formal definition is given of what a variation is, and thus I have some questions:
- A functional seems to be defined as a map $C\to\mathbb{R}$ where $C$ is the "space of curves". How exactly is $C$ defined?
- Is $C$ a normed vector space? If so, can differentiability of a functional be stated in terms of the total derivative?
