The main points are:

1. We are studying a Canonical Transformation (CT) $$(q,p) \longrightarrow (Q,P) $$ from old canonical coordinates $(q,p)$ and Hamiltonian $H(q,p,t)$ to new canonical coordinates $(Q,P)$and Kamiltonian $K(Q,P,t)$.

2. $S_1(q,Q,t)$ is a so-called type 1 generating function of the CT.

3. $S_2(q,P,t)$ is a so-called type 2 generating function of the CT.

4. For all four types of generating functions hold that $$K-H~=~\frac{\partial S_i}{\partial t},\qquad i~=~1,2,3,4. $$

5. The two types of generating function are connected via a Legendre transformation $$S_2(q,P,t)-S_1(q,Q,t)~=~Q^i P_i. $$

6. Goldstein, Classical Mechanics, uses $S_2(q,P,t)$ in the treatment of Hamilton-Jacobi equation. Goldstein assumes that the Kamiltonian $K=0$ vanishes identically.

7. Arnold, Mathematical Methods of Classical Mechanics, uses $S_1(q,Q,t)$ in Section 47 and $S_2(q,P,t)$ in Section 48. Arnold assumes (among other things) that $S_1(q,Q,t)$ does not depend explicitly on $t$.

The main points are:

1. We are studying a Canonical Transformation (CT) $$(q,p) \longrightarrow (Q,P) $$ from old canonical coordinates $(q,p)$ and Hamiltonian $H(q,p,t)$ to new canonical coordinates $(Q,P)$and Kamiltonian $K(Q,P,t)$.

2. $S_1(q,Q,t)$ is a so-called type 1 generating function of the CT.

3. $S_2(q,P,t)$ is a so-called type 2 generating function of the CT.

4. The two types of generating function are connected via a Legendre transformation $$S_2(q,P,t)-S_1(q,Q,t)~=~Q^i P_i. $$

5. For all four types of generating functions hold that $$K-H~=~\frac{\partial S_i}{\partial t},\qquad i~=~1,2,3,4. $$

6. Goldstein, Classical Mechanics, uses $S_2(q,P,t)$ in the treatment of Hamilton-Jacobi equation. Goldstein assumes that the Kamiltonian $K=0$ vanishes identically.

7. Arnold, Mathematical Methods of Classical Mechanics, uses $S_1(q,Q,t)$ in Section 47 and $S_2(q,P,t)$ in Section 48. Arnold assumes (among other things) that $S_1(q,Q,t)$ does not depend explicitly on $t$.

The main points are:

1. We are studying a Canonical Transformation (CT) $$(q,p) \longrightarrow (Q,P) $$ from old canonical coordinates $(q,p)$ to new canonical coordinates $(Q,P)$.

2. $S_1(q,Q,t)$ is a so-called type 1 generating function of the CT.

3. $S_2(q,P,t)$ is a so-called type 2 generating function of the CT.

4. The two types of generating function are connected via a Legendre transformation $$S_2(q,P,t)-S_1(q,Q,t)~=~Q^i P_i. $$

The main points are:

1. We are studying a Canonical Transformation (CT) $$(q,p) \longrightarrow (Q,P) $$ from old canonical coordinates $(q,p)$ and Hamiltonian $H(q,p,t)$ to new canonical coordinates $(Q,P)$and Kamiltonian $K(Q,P,t)$.

2. $S_1(q,Q,t)$ is a so-called type 1 generating function of the CT.

3. $S_2(q,P,t)$ is a so-called type 2 generating function of the CT.

4. For all four types of generating functions hold that $$K-H~=~\frac{\partial S_i}{\partial t},\qquad i~=~1,2,3,4. $$

5. The two types of generating function are connected via a Legendre transformation $$S_2(q,P,t)-S_1(q,Q,t)~=~Q^i P_i. $$

6. Goldstein, Classical Mechanics, uses $S_2(q,P,t)$ in the treatment of Hamilton-Jacobi equation. Goldstein assumes that the Kamiltonian $K=0$ vanishes identically.

7. Arnold, Mathematical Methods of Classical Mechanics, uses $S_1(q,Q,t)$ in Section 47 and $S_2(q,P,t)$ in Section 48. Arnold assumes (among other things) that $S_1(q,Q,t)$ does not depend explicitly on $t$.