How does Lagrangian turn into Hamiltonian?

 In Section 6 of Feynman’s 1948 paper, he proved the equivalence of the path integral approach to Schrodinger’s equation. Path integral is based on Lagrangian. Schrodinger’s equation is based on Hamiltonian. How does Lagrangian turn into Hamiltonian?

At the technical level, i, the imaginary number does the trick. Square the imaginary number and you get negative one. But at the conceptual level, how can Lagrangian be equivalent to Hamiltonian?

P.S. Schrodinger equation, a PDE, is equivalent to path integral. Feynman Kac formula, a PDE, is also equivalent to path integral. What are their differences from methodology perspective?