# Symplectic Geometry of Constrained Optimization

## TL;DR

This paper explores how symplectic geometry, especially the Maslov index, can be used to compute the Morse index in constrained optimization problems, even when they are highly degenerate.

## Contribution

It introduces a geometric approach using symplectic geometry and the Maslov index to analyze the second variation in constrained optimization, simplifying complex calculations.

## Key findings

- Effective computation of Morse index in degenerate problems
- Application of symplectic geometry to constrained optimization
- Use of Maslov index as a key analytical tool

## Abstract

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.