Seonglae ChoIn 1837, Gustav Jacobi introduced Canonical Transformation in generalized coordinates and generalized momenta to simplify the Hamiltonian and make it easier to find solutions to Hamilton's equations. He described the relationships between variables as partial differential equations, which are generating functions that appear due to the Gauge Invariance of the Lagrangian.
Hamilton–Jacobi equation
In physics, the Hamilton–Jacobi equation, named after William Rowan Hamilton and Carl Gustav Jacob Jacobi, is an alternative formulation of classical mechanics, equivalent to other formulations such as Newton's laws of motion, Lagrangian mechanics and Hamiltonian mechanics.
https://en.wikipedia.org/wiki/Hamilton–Jacobi_equation