Does anybody know of a short lecture script/website/small book that gives an introduction to the calculus of variations?
- It should be suitable for total beginners, with steps explained in detail.
- At the moment, I'm not looking for proofs or advanced topics, just the basics, more some kind of cheat sheet than a real introduction book.
I'm mostly interested in the calculus of variations that's being used in Lagrange and Hamiltonian formalisms. I have physics scripts that do things like:
$$ \delta S = \int_{t_1}^{t_2}dt \frac{\delta S}{\delta \textbf{q}(t)} [\textbf{Q}]\delta\textbf{q}(t) $$
for the action $S[\textbf{q}]$ in classical Lagrange formalism (with $\textbf{q}$ the path of the particle and $\textbf{Q}$ the path with the minimal action).
I'd like to have a basic understanding of such derivations.