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I want to understand the relation between several different definitions of canonical transformation. I am studying the answer by Qmechanic in this post

Let us define a canonical transformation as a transformation that satisfies $(\sum_ip_idq^i-Hdt)-(\sum_iP_i dQ^i-Kdt)=dF$ for some generating function $F$. Any possible time-dependence is allowed.

Also, let us define a symplectomorphism as a transformation that preserves the symplectic two-form, i.e., $f^*\omega=\omega$ where $\omega=\sum_idp_i\wedge dq^i$ in local Darboux coordinate.

According to the answer, a canonical transformation is a symplectomorphism, but a symplectomorphism is not generally a canonical transformation as “there may be global obstructions”. Could anyone give us a concrete example of such a global obstruction? I guess the global topology of $M$ is important. I am looking for an example of symplectomorphism for a phase space $M$ that is relevant for classical mechanics, which is not a canonical transformation.

I looked at a related post and the answer but it does not give us an example.

Also, H. Goldstein, Classical Mechanics uses the same definition of the canonical transformation and states in Sec 9.4 that “the symplectic condition remains a necessary and sufficient condition for a canonical transformation even if it involves the time”. Since the symplectic condition should be equivalent to the invariance of the symplectic two-form ($\sum_idP_i\wedge dQ^i=\sum_idp_i\wedge dq^i$ in Darboux coordinate), this book seems to imply that any symplectomorphism is a canonical transformation. This point is mentioned by this post but again the answer was not concrete.

Qmechanic
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watahoo
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    Related: https://physics.stackexchange.com/q/391216/2451 – Qmechanic Jan 09 '24 at 07:33
  • So, can the following be a possible concrete example?

    Let us consider the case where the phase space is a 2-torus $S^1\times S^1$. Let $(q,p)$ be a coordinate. $q+1$ and $p+1$ are identified with $q$ and $p$. I assume $H=K$ is invariant under $q\to q+1$ and $p\to p+1$.

    The map $Q(q,p)=q+np$, $P(q,p)=p$ is a symplectomorphism, because it preserves the symplectic two-form. However, there is no globally defined $F$ unless $n=0$, since $(pdq-Hdt)-(QdP-Kdt)=dF$ with$H=K$ is satisfied by the choice $F=-\frac{1}{2}np^2$ but this is not invariant under $p\to p+1$.

    Any commet will be helpful.

     – watahoo Jan 10 '24 at 04:30

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