Generating function of a canonical transformation

Definition

Let $n \ge 1$ and let $(q, p)$ and $(Q, P)$ be old and new phase-space variables in $\R^{2n}$, with $t$ the time.

1. A generating function of type 1 is a function $F_1(q, Q, t)$ of class $C^2$ such that the matrix $\big(\partial^2 F_1 / \partial q_i \partial Q_j\big)$ is invertible. It generates the transformation $(q, p) \mapsto (Q, P)$ defined implicitly by $$p_i = \frac{\partial F_1}{\partial q_i}(q, Q, t), \qquad P_i = -\frac{\partial F_1}{\partial Q_i}(q, Q, t).$$
2. A generating function of type 2 is a function $F_2(q, P, t)$ of class $C^2$ such that the matrix $\big(\partial^2 F_2 / \partial q_i \partial P_j\big)$ is invertible. It generates the transformation defined implicitly by $$p_i = \frac{\partial F_2}{\partial q_i}(q, P, t), \qquad Q_i = \frac{\partial F_2}{\partial P_i}(q, P, t).$$

In both cases, the new Hamiltonian is $K = H + \partial F / \partial t$, where $F$ is the generating function and $H$ is the old Hamiltonian.