- #1

andresB

- 626

- 374

$$(q_{0},p_{0})\rightarrow(q_{\lambda},p_{\lambda})$$

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 ##??