§ Canonical Transformations
The content of this post is based on the
notes
written by
Giancarlo Benettin. These are not only exceptionally well
written, but are also very rigorous in their approach. I will try to summarize briefly the most important concepts to clarify
what
canonical transformations are.
Definition I. Let \(\Omega\subseteq \mathbb{R}^{2n}\) and \(\tilde{\Omega}\subseteq \mathbb{R}^{2n}\)
be two open subsets. Consider coordinates \((\mathbf{p},\mathbf{q})\) on \(\Omega\) and \((\tilde{\mathbf{p}},\tilde{\mathbf{q}})\)
on \(\tilde{\Omega}\). Let
$$
\begin{align}
w\colon \tilde{\Omega} &\to \Omega\\
(\tilde{\mathbf{p}},\tilde{\mathbf{q}}) &\mapsto w(\tilde{\mathbf{p}},\tilde{\mathbf{q}}):= (u(\tilde{\mathbf{p}},\tilde{\mathbf{q}}), v(\tilde{\mathbf{p}},\tilde{\mathbf{q}}))
\end{align}
$$
be a diffeomorphism (differentiable map whose inverse is also differentiable). We will say that \(w\) defines a
restricted time-independent canonical transformation if, for any \(H\) defining a Hamiltonian system
in \(\Omega\), the function \(\tilde{H}:=H\circ w\) defines a hamiltonian system in \(\tilde{\Omega}\), i.e.
$$
\dot{\tilde{\mathbf{p}}}= - \dfrac{\partial \tilde{H}}{\partial \tilde{\mathbf{q}}} \quad \quad \dot{\tilde{\mathbf{q}}}= \dfrac{\partial \tilde{H}}{\partial \tilde{\mathbf{p}}}
$$
Note that, abusing (a bit) of notation, we can view the coordinates \(p,q\) as functions of \((\tilde{\mathbf{p}},\tilde{\mathbf{q}})\),
this meaning that we may call \(u=p\), and \(v=q\).
Example. Let's consider \(\Omega = \tilde{\Omega}=\mathbb{R}^2\) and for \(a,b\in\mathbb{R}\) the diffeomorphism
$$
\begin{align}
w\colon \tilde{\Omega} &\to \Omega\\
(\tilde{p},\tilde{q}) &\mapsto (u(\tilde{p},\tilde{q}), v(\tilde{p},\tilde{q})):=(\tilde{p}+a,\tilde{q}+b)
\end{align}
$$
In order to see that \(w\) defines a restricted time-independent canonical transformation, we need to
verify that, for any Hamiltonian \(H\) that defines the Hamiltonian system
$$
\dot{p} = -\dfrac{\partial H}{\partial q} \quad \dot{q} = \dfrac{\partial H}{\partial p}
$$
the fucntion \(\tilde{H}:= H\circ w\) also defines a Hamiltonian system, but in the new variables, namely that
$$
\dot{\tilde{p}} = -\dfrac{\partial \tilde{H}}{\partial \tilde{q}} \quad \dot{\tilde{q}} = \dfrac{\partial \tilde{H}}{\partial \tilde{p}}
$$
Indeed, recalling the abuse of notation \(p=u\), we have:
$$
\begin{align}
\dfrac{\partial \tilde{H}}{\partial \tilde{q}} &= \dfrac{\partial}{\partial \tilde{q}}(H\circ w)=
\begin{pmatrix} \dfrac{\partial H}{\partial p} & \dfrac{\partial H}{\partial q}\end{pmatrix}
\begin{pmatrix} \dfrac{\partial p}{\partial \tilde{q}} \\ \dfrac{\partial q}{\partial \tilde{q}}\end{pmatrix}=
\begin{pmatrix} \dot{q} & -\dot{p}\end{pmatrix} \begin{pmatrix} 0 \\ 1\end{pmatrix}=\\
&= -\dot{p} = -\dot{\tilde{p}+a}= -\dot{\tilde{p}}
\end{align}
$$
Thus, we get the desired result
$$
\dot{\tilde{p}} = -\dfrac{\partial \tilde{H}}{\partial \tilde{q}}
$$
Similarly one can also check that the result holds for for the other Hamilton equation.
Definition II. Let \(\Omega\subseteq \mathbb{R}^{2n}\) and \(\tilde{\Omega}\subseteq \mathbb{R}^{2n}\)
be two open subsets. Consider coordinates \((\mathbf{p},\mathbf{q})\) on \(\Omega\) and \((\tilde{\mathbf{p}},\tilde{\mathbf{q}})\)
on \(\tilde{\Omega}\). Let
$$
\begin{align}
w\colon \tilde{\Omega} &\to \Omega\\
(\tilde{\mathbf{p}},\tilde{\mathbf{q}}) &\mapsto w(\tilde{\mathbf{p}},\tilde{\mathbf{q}}):= (u(\tilde{\mathbf{p}},\tilde{\mathbf{q}}), v(\tilde{\mathbf{p}},\tilde{\mathbf{q}}))
\end{align}
$$
be a diffeomorphism (differentiable map whose inverse is also differentiable). We will say that \(w\) defines a
general time-independent canonical transformation if, for any \(H\) defining a Hamiltonian system
in \(\Omega\), there exists a function \(\tilde{H}\colon \tilde{\Omega}\to\mathbb{R}\) that defines a hamiltonian
system in \(\tilde{\Omega}\), i.e.
$$
\dot{\tilde{\mathbf{p}}}= - \dfrac{\partial \tilde{H}}{\partial \tilde{\mathbf{q}}} \quad \quad \dot{\tilde{\mathbf{q}}}= \dfrac{\partial \tilde{H}}{\partial \tilde{\mathbf{p}}}
$$
Note that this differs from
Definition I only by the fact that here, no special definition of \(\tilde{H}\)
is given. This makes
Definition II more general than
Definition I. In particular, this means that
there might exist transformations that are general time-independent canonical that are not restricted time-independent canonical. For
instance, in the case \(\Omega = \tilde{\Omega}=\mathbb{R}^2\) and \(a,b\in\mathbb{R}\) verifying \(a\cdot b= c^{-1}\)
$$
\begin{align}
w\colon \tilde{\Omega} &\to \Omega\\
(\tilde{p},\tilde{q}) &\mapsto (u(\tilde{p},\tilde{q}), v(\tilde{p},\tilde{q})):= (a\tilde{p}, b\tilde{q})
\end{align}
$$
is general time-independent canonical when \(\tilde{H}=cH\circ w\), which is (generally) different from \(H\circ w\).
It is also common to define canonical transformations as those who preserve the Poisson brackets. We now make clear what we mean
by this.
Definition III. Let \(\Omega\subseteq \mathbb{R}^{2n}\) and \(\tilde{\Omega}\subseteq \mathbb{R}^{2n}\)
be two open subsets. Consider coordinates \((\mathbf{p},\mathbf{q})\) on \(\Omega\) and \((\tilde{\mathbf{p}},\tilde{\mathbf{q}})\)
on \(\tilde{\Omega}\). Let
$$
\begin{align}
w\colon \tilde{\Omega} &\to \Omega\\
(\tilde{\mathbf{p}},\tilde{\mathbf{q}}) &\mapsto w(\tilde{\mathbf{p}},\tilde{\mathbf{q}}):= (u(\tilde{\mathbf{p}},\tilde{\mathbf{q}}), v(\tilde{\mathbf{p}},\tilde{\mathbf{q}}))
\end{align}
$$
be a diffeomorphism (differentiable map whose inverse is also differentiable). Furthermore, let
$$
f,g\colon \Omega \to \mathbb{R}
$$
be two differentiable functions. We will say that \(w\) preserves the Poisson bracket \(\{f,g\}\) if
$$
\{f\circ w, g\circ w\}_{(\tilde{\mathbf{p}}, \tilde{\mathbf{q}})} = \{f, g\}_{(\mathbf{p}, \mathbf{q})}\circ w
$$
Proposition IV. \(w\) preserves the Poisson bracket if and only if it is a restricted time-independent
canonical transformation.
Proposition IV. \(w\) preserves the Poisson bracket if and only if
$$
\{u_i, u_j\}_{(\tilde{\mathbf{p}}, \tilde{\mathbf{q}})} = 0 = \{v_i, v_j\}_{(\tilde{\mathbf{p}}, \tilde{\mathbf{q}})}
\quad \{v_i, u_j\}_{(\tilde{\mathbf{p}}, \tilde{\mathbf{q}})}= \delta_{ij} \quad \forall i,j = 1,\ldots, n
$$
For generalized time-independent canonical transformations, the preservation of the Poisson brackets takes a different form.