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Consider the phase space of a one degree of freedom mechanical system. We can pass from one phase space coordinates to another phase space coordinates via a canonical transformation. I want to focus on 1-parameter canonical transformations,
$$(q_{0},p_{0})\rightarrow(q_{\lambda},p_{\lambda})$$
where ##\lambda\in[0,\infty)## parametrize the transformation.
By the standard theory, there exist a function ##F=F_{1}(q_{0},q_{\lambda};\lambda)## such that
$$p_{0}\frac{dq_{0}}{dt}-H=p_{\lambda}\frac{dq_{\lambda}}{dt}-K+\frac{dF_{1}}{dt}.$$
##F## is called the generator of the transformation, and the following equation follows
$$p_{0} =\frac{\partial F_{1}}{\partial q_{0}},\qquad p_{\lambda}=-\frac{\partial F_{1}}{\partial q_{\lambda}}.$$
Now, also by standard theory, there exist a function ##W=W(q,p;\lambda)## such that the transformation can be obtained via the Poisson brackets using the equations
$$\frac{dq}{d\lambda} =\left\{ q,W\right\}, $$
$$\frac{dp}{d\lambda} =\left\{ p,W\right\}.$$
##W## is again sometimes called the generator of the transformation.
What is the relation between ##F## and ##W ##??
$$(q_{0},p_{0})\rightarrow(q_{\lambda},p_{\lambda})$$
where ##\lambda\in[0,\infty)## parametrize the transformation.
By the standard theory, there exist a function ##F=F_{1}(q_{0},q_{\lambda};\lambda)## such that
$$p_{0}\frac{dq_{0}}{dt}-H=p_{\lambda}\frac{dq_{\lambda}}{dt}-K+\frac{dF_{1}}{dt}.$$
##F## is called the generator of the transformation, and the following equation follows
$$p_{0} =\frac{\partial F_{1}}{\partial q_{0}},\qquad p_{\lambda}=-\frac{\partial F_{1}}{\partial q_{\lambda}}.$$
Now, also by standard theory, there exist a function ##W=W(q,p;\lambda)## such that the transformation can be obtained via the Poisson brackets using the equations
$$\frac{dq}{d\lambda} =\left\{ q,W\right\}, $$
$$\frac{dp}{d\lambda} =\left\{ p,W\right\}.$$
##W## is again sometimes called the generator of the transformation.
What is the relation between ##F## and ##W ##??