The time evolution governed by Hamilton's equations is an example of continuous canonical transformation. The infinitesimal generator of this transformation is the Hamiltonian itself.
In the phase space formulation, the constant of motion \(G\) given by Noether's theorem, expressed in terms of coordinates and momenta generates the infinitesimal symmetry transformation.
$\newcommand{\pp}[2][]{\frac{\partial#1}{\partial #2}}$
The definition of infnitesimal transformations is given.
The set of all canonical transformations given by a generator of one of the four types form a group.
The transformations
form important canonical transformations.
$\newcommand{\pp}[2][]{\frac{\partial#1}{\partial #2}} \newcommand{\dd}[2][]{\frac{d#1}{d #2}} \newcommand{\DD}[2][]{\frac{d^2#1}{d #2^2}} \newcommand{\Label}[1]{\label{#1}} \newcommand{\EqRef}[1]{Eq.\eqref{#1}} \newcommand{\eqRef}[1]{\eqref{#1}}$
Canonical transformation is defined in three different,equivalent ways.
$\newcommand{\pp}[2][]{\frac{\partial #1}{\partial #2}}$A canonical transformation is a change of variables \((q,p) \rightarrow (Q,P)\) in phase space such that the Hamiltonian form of equations of motion is preserved. Depending choice of independent variables we have four special cases of canonical transformations., Generating functions for the four cases are introduced and details of the four cases are discussed.
Important relations of four types of transformations are summarized.
Several examples on canonical transformations are given.
The definition finite and infinitesimal canonical transformation are given. Using the action principle we define the generator of a canonical transformation.$\newcommand{\pp}[2][]{\frac{\partial#1}{\partial #2}}\newcommand{\Label}[1]{\label{#1}}$
$\newcommand{\pp}[2][]{\frac{\partial#1}{\partial #2}} \newcommand{\PP}[2][]{\frac{\partial^2#1}{\partial #2^2}} \newcommand{\dd}[2][]{\frac{d#1}{d #2}}\newcommand{\DD}[2][]{\frac{d^2#1}{d #2^2}}\newcommand{\Label}[1]{\label{#1}}\newcommand{\eqRef}[1]{\eqref[#1]}$A canonical transformation is a change of variable in phase space such that the equations of motion in the new variables are of the Hamiltonian form.
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