Derivation of Hamilton-Jacobi theory using canonical transformations

Question

The derivation of the Hamilton-Jacobi equation using canonical transformations is typically done involving a type-2 generating function.

Is it possible to use a another type of generating function, namely type 1, 3 or 4?

Answer 1

Yes! We definitely can. However, it will not make a difference.

The idea with the Hamilton-Jacobi Equation is that we're looking at not just any generator, but specifically the generator that brings us to the coordinates $\{Q, P\}$ such that the new Hamiltonian $K$ ("Kamiltonian", as Goldstein refers to it) is now 0. Thus: $$\frac{\partial K}{\partial Q_i} = - \dot{P_i} = 0 \Rightarrow P_i = \alpha_i$$ $$\frac{\partial K}{\partial P_i} = \dot{Q_i} = 0 \Rightarrow Q_i = \beta_i$$

So the new coordinates are constants of motion. This means that the action will always be in terms of the old coordinates and momenta, no matter how we define it. Action as a convention however is almost always written in terms of coordinates, as it is given by the integral of the Lagrangian.

Hope that helps!