I The different generators of canonical transformations

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The discussion focuses on 1-parameter canonical transformations in the phase space of a one degree of freedom mechanical system, emphasizing the relationship between two generators of transformation: F and W. It highlights that F is defined through a specific function relating phase space coordinates and their derivatives, while W can be derived using Poisson brackets. A participant expresses difficulty finding references on this topic in standard mechanics texts and requests online resources. Another contributor mentions their understanding comes from Goldstein's mechanics textbook and references an old paper that generalizes the relationship between Hamilton's principal functions and the Hamiltonian. The conversation underscores the complexity of canonical transformations and the need for accessible resources.
andresB
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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 ##??
 
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andresB said:
Now, also by standard theory, there exist a function W=W(q,p;λ) such that the transformation can be obtained via the Poisson brackets using the equations
As a layman I do not find this transformation via the Poisson brackets in my text of mechanics. Could you show me some web reference if you know any ?
 
anuttarasammyak said:
As a layman I do not find this transformation via the Poisson brackets in my text of mechanics. Could you show me some web reference if you know any ?
I don't know any good online references. My knowledge of the topics comes from Goldstein 2nd edition, section 9.5.
 
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