# The different generators of canonical transformations

• I
• andresB
In summary, the conversation discusses 1-parameter canonical transformations and the existence of generators for these transformations. The generators, ##F## and ##W##, have different definitions but are related through the Poisson brackets. There is a lack of online references for this topic, but it can be found in Goldstein's book "Classical Mechanics" in section 9.5.
andresB
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 ##??

vanhees71
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.

anuttarasammyak

## 1. What is a canonical transformation?

A canonical transformation is a mathematical transformation that preserves the fundamental structure of a physical system, such as Hamilton's equations of motion and the Poisson bracket. It allows us to change the coordinates and momenta of a system while maintaining its underlying dynamics.

## 2. What are the generators of canonical transformations?

The generators of canonical transformations are functions of the old coordinates and momenta that determine the new coordinates and momenta after the transformation. They can be either first-class or second-class, depending on whether they commute or not with the original Hamiltonian.

## 3. How do the generators of canonical transformations relate to symmetries?

The generators of canonical transformations are closely related to symmetries in a physical system. In fact, Noether's theorem states that every continuous symmetry of a system corresponds to a conserved quantity, which can be expressed as a generator of canonical transformations.

## 4. Can canonical transformations change the form of Hamilton's equations?

No, canonical transformations do not change the form of Hamilton's equations. They only change the coordinates and momenta of a system, while preserving the underlying dynamics described by Hamilton's equations.

## 5. How are canonical transformations useful in physics?

Canonical transformations are useful in physics because they allow us to simplify the mathematical description of a physical system by transforming it into a more convenient set of coordinates and momenta. They also help us identify symmetries and conserved quantities in a system, which can provide valuable insights into its behavior and dynamics.

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