  These are the notes of rather informal lectures given by the first co-author
in UPMC, Paris, in January 2017. Practical goal is to explain how to compute or
estimate the Morse index of the second variation. Symplectic geometry allows to
effectively do it even for very degenerate problems with complicated
constraints. Main geometric and analytic tool is the appropriately rearranged
Maslov index.
  In these lectures, we try to emphasize geometric structure and omit analytic
routine. Proofs are often substituted by informal explanations but a
well-trained mathematician will easily re-write them in a conventional way.
