Connection between Lagrangian and (pre)symplectic structure

Question

How is the precise relationship between a Lagrangian and the (pre)symplectic structure?

Given a (pre)symplectic structure can I construct a (equivalence class of) Lagrangian(s)?

References are very welcome.

Answer 1

1. Given a Lagrangian $L$, it is in principle possible to perform a (possible singular) Legendre transformation using e.g. the Faddeev-Jackiw method to arrive at a Hamiltonian Lagrangian of the form $$L_H(z,\dot{z},t)~=~\vartheta_I(z,t)~\dot{z}^I - H(z,t)\tag{6}$$ with a Hamiltonian $H$ and a pre-symplectic potential $$\vartheta~=~\vartheta_I(z,t)~\mathrm{d}z^I, \qquad \omega~:=~\mathrm{d}\vartheta. $$

2. Conversely, given a Hamiltonian $H$ and a pre-symplectic potential $\vartheta$, the Hamiltonian Lagrangian (6) provides a Lagrangian formulation.