Question about canonical transformation
Cool question!
Thanks to user lionelbrits for his answer that prompted me to pull out my mechanics books and check the definitions of "canonical transformation" given by different authors.
If you look in Goldstein's classical mechanics texts in the section on canonical transformations, then you'll find that canonical transformations are essentially defined as follows (I paraphrase)
Goldstein Definition: A transformation $f:\mathcal P\to\mathcal P$ on phase space $\mathcal P$ is canonical provided there exists a phase space function $K$ such that if $(q(t), p(t))$ is a solution to Hamilton's equations generated by $H$, then $(Q(t), P(t)) = f(q(t), p(t))$ is a solution to Hamilton's equations generated by $K$.
This is essentially the definition given by lionelbrits in his answer.
On the other hand, if you look, for example, in Spivak's mechanics text, then you'll find the following definition:
Spivak's Definition: A transformation $f:\mathcal P \to \mathcal P$ on phase space is canonical provided it preserves the symplectic form.
In more concrete terms (namely in canonical coordinates), Spivak's definition can be stated as follows:
The transformation $f(q,p) = (f^q(q,p), f^p(q,p))$ is canonical if and only if its Jacobian (derivative) matrix preserves the symplectic matrix $J$, namely \begin{align} f'(p,q)\,J\,f'(p,q)^t = J \end{align} where \begin{align} J=\begin{pmatrix} 0 & I_n \\ -I_n & 0 \\ \end{pmatrix},\qquad f' = \begin{pmatrix} \frac{\partial f^q}{\partial q} & \frac{\partial f^q}{\partial p} \\ \frac{\partial f^p}{\partial q} & \frac{\partial f^p}{\partial p} \\ \end{pmatrix} \end{align} where $2n$ is the dimension of phase space and $I_n$ is the $n\times n$ identity matrix.
It also turns out that
If a transformation is canonical in the sense defined by Spivak, then it is canonical is the sense of Goldstein with $K = H\circ f^{-1}$
but the converse is not true. In fact, this example you brink up is a counterexample to the converse! What lionelbrit showed in his answer is that the example you have written is a canonical transformation in the sense of Goldstein, but, as you should try to convince yourself (I did), the function $K = H\circ f^{-1}$ that you wrote down by inverting the transformation and plugging back into $H$ leads to Hamilton's equations that are not satisfied by $(Q(t), P(t)) = f(q(t), p(t))$. You can show this directly by writing down the equations of motion. You can also show this by computing the Jacobian of the transformation and showing that it does not preserve the symplectic matrix. In fact, you should find that the Jacobian is given by \begin{align} f'(q,p)=\begin{pmatrix} 1 & 0 \\ -\frac{1}{2\sqrt{q}} & \frac{1}{2\sqrt{p}} \\ \end{pmatrix} \end{align} and that \begin{align} f'(q,p) J f'(q,p)^t = \frac{1}{2\sqrt{p}} J \end{align} In other words, the Jacobian of the transformation preserves the symplectic matrix up to a multiplicative factor.
Speculation. I'm going to go out on a limb and guess that your professor calls Goldstein's definition a "local canonical transformation" and Spivak's definition a "canonical transformation." If we adopt this terminology, then it's clear from our remarks that the $K$ he gives shows that your example is a local canonical transformation, but that the transformation is not canonical.
I) The restricted$^1$ transformation (RT)
$$ (q,p)~\longrightarrow~ (Q,P) ~:=~(q, \sqrt{p} - \sqrt{q})\tag{1}$$
of OP's professor with inverse RT
$$ (Q,P)~\longrightarrow~ (q,p) ~:=~(Q, (P+ \sqrt{Q})^2) ,\tag{2}$$
and with Hamiltonian $H=\frac{p^2}{2}$ and Kamiltonian $K=\frac{p^{3/2}}{3}$ is indeed interesting. OP's example (1) is mentioned in Ref. 3. Apparently we should assume that $p,q,Q\geq 0$ and $P+\sqrt{Q}\geq 0$.
II) As joshphysics essentially writes in his answer, the RT (1) is not a symplectomorphism, because the Poisson bracket is not preserved if $p\neq \frac{1}{4}$:
$$ \{Q,P\} ~=~\frac{\{q,p\}}{2\sqrt{p}}~\neq~\{q,p\}~=~1. \tag{3}$$
III) As lionelbrits shows in his answer, the RT (1) do transform the Hamilton's eqs. into Kamilton's eqs, which according to Wikipedia (December 2013) is the defining property of a canonical transformation (CT). Goldstein, Landau and Lifshitz (Ref. 1 and 2) disagree with such a definition of CT. Ref. 1 and 2 state that form invariance is only a necessary but not a sufficient condition for being a canonical transformation (CT). Ref. 3 calls the transformation (1) a canonoid transformation. See also this related Phys.SE post.
IV) Both Refs. 1 and 2 define a CT as satisfying
$$ (p\dot{q}-H)-(P\dot{Q}-K)~=~\frac{dF}{dt},\tag{4}$$
or equivalently
$$ (p\mathrm{d}q-H\mathrm{d}t)-(P\mathrm{d}Q -K\mathrm{d}t) ~=~\mathrm{d}F,\tag{5} $$
for some function $F$. Or equivalently (ignoring possible topological obstructions),
$$ \mathrm{d}\left(p\mathrm{d}q-P\mathrm{d}Q +(K-H)\mathrm{d}t\right)~=~0. \tag{6}$$
For OP's example the condition (6) does not hold
$$ \tag{7} \mathrm{d}\left((p-\sqrt{p}+\sqrt{q})\mathrm{d}q +(\frac{p^{3/2}}{3}-\frac{p^2}{2})\mathrm{d}t\right) ~\neq~0. $$
So OP's example is not a CT according to Refs. 1 and 2.
References:
H. Goldstein, Classical Mechanics, Chapter 9. See text under eq. (9.11).
L.D. Landau and E.M. Lifshitz, Mechanics, $\S45$. See text between eqs. (45.5-6).
J.V. Jose & E.J. Saletan, Classical Dynamics: A Contemporary Approach, 1998; Subsection 5.3.1, p. 233.
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The word restricted means that the transformation $(q,p)\longrightarrow (Q,P)$ has no explicit time dependence.
The original coordinates satisfy the equations of motion when the integral of $p\, \dot{q} - H(p,q)$ is minimized, and the new coordinates satisfy the equations of motion when the integral of $P\, \dot{Q} - K(P,Q)$ is minimized. There is no requirement that $H$ and $K$ be numerically equal.
The transformation is canonical if the Poisson bracket remains invariant.
The EOMs are
$\dot{p} = 0$
$\dot{q} = p$
and from the new Hamiltonian, we get
$\dot{P} = -(P+\sqrt{Q})^2 \frac{1}{2\sqrt{Q}} = - \frac{p}{2\sqrt{q}} = \frac{d}{dt} \left(\sqrt{p} - \sqrt{q} \right)$
$\dot{Q} = (P+\sqrt{Q})^2 = \dot{q}$
thus the equations of motion are numerically equal.