In Goldstein's mechanics book he defines a canonical transformation as a transformation from $q, p$ to $Q, P$ with $$ Q = Q(q,p,t) \tag{9.4a} $$ and $$ P = P(q,p,t) \tag{9.4b} $$ such that if $H$ is the Hamiltonian then there exists a $K$ and $F$ such that $$ p_i \dot{q}_i - H = P_i \dot{Q}_i - K+ \frac{dF}{dt} \tag{9.11}. $$ Later he shows that a time dependent canonical transformation satisfies the symplectic condition. He starts with a transformation of the form $$ \boldsymbol{\zeta} = \boldsymbol{\zeta}(\boldsymbol{\eta}, t)\tag{9.59}$$ which is assumed to be canonical. Then there are two statements. The first is that the transformation $$\boldsymbol{\eta} \to \boldsymbol{\zeta}(t_0)\tag{9.60b}$$ is ''obviously'' canonical. The second is that the transformation $$\boldsymbol{\zeta}(t_0) \to \boldsymbol{\zeta}(t)\tag{9.60c}$$ is also canonical.

Now although these statements are intuitive, they do not seem obvious to me. With the first statement, it appears you must construct and $K$ and $F$ for the transformation to $\boldsymbol{\zeta}( t_0)$ assuming these exist for the general time-dependent transformation. Would someone be able to help with my difficulties?


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