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  1. Which transformations are canonical?

  2. Why do canonical transformations preserve the measure of integration in phase space?

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Q1) Be aware that there exist various definitions of a canonical transformation (CT) in the literature:

  1. Firstly, Refs. 1 and 2 define a CT as a transformation$^1$ $$\tag{1} (q^i,p_i)~~\mapsto~~ \left(Q^i(q,p,t),P_i(q,p,t)\right)$$ [together with choices of a Hamiltonian $H(q,p,t)$ and a Kamiltonian $K(Q,P,t)$; and where $t$ is the time parameter] that satisfies $$ \tag{2} (p_i\mathrm{d}q^i-H\mathrm{d}t) -(P_i\mathrm{d}Q^i -K\mathrm{d}t) ~=~\mathrm{d}F$$ for some generating function $F$.

  2. Secondly, Wikipedia (October 2015) calls a transformation (1) a CT if it transforms the Hamilton's eqs. into Kamilton's eqs.

  3. Thirdly, some authors (e.g. Ref. 3) use the word CT as just another word for a symplectomorphism $f:M\to M$ [which may depend on a parameter $t$] on a symplectic manifold $(M,\omega)$, i.e. $$\tag{3} f^{\ast}\omega=\omega.$$ Here $\omega$ is the symplectic two-form, which in local Darboux/canonical coordinates reads $\omega= \mathrm{d}p_i\wedge \mathrm{d}q^i$.

  4. Fourthly, Ref. 1 defines an extended canonical transformation (ECT) as a transformation (1) [together with choices of a Hamiltonian $H(q,p,t)$ and a Kamiltonian $K(Q,P,t)$; and where $t$ is the time parameter] that satisfies $$ \tag{4} \lambda(p_i\mathrm{d}q^i-H\mathrm{d}t) -(P_i\mathrm{d}Q^i -K\mathrm{d}t) ~=~\mathrm{d}F$$ for some parameter $\lambda\notin \{0\}$ and for some generating function $F$.

Now let us discuss the relationship between the above four different definitions.

  1. The first definition is an ECT with $\lambda=1$. An ECT satisfies the second definition, but not necessarily vice-versa, cf. e.g. this and this Phys.SE post.

  2. The first definition is a symplectomorphism (by forgetting about $H$ and $K$). Conversely, there may be global obstructions for a symplectomorphism to satisfy the first definition. However, a symplectomorphism sufficiently close to the identity map and defined within a single Darboux coordinate chart does satisfy the parts of the first definition that do not concern $H$ and $K$. See also e.g. my Phys.SE answer here.

  3. An ECT is not necessarily a symplectomorphism. Counterexample: $$\tag{5} Q~=~\lambda q, \qquad P=p \qquad K~=~\lambda H, \qquad F~=~0,$$ where $\lambda\notin \{0,1\}$ is a constant different from zero and one, so that the Poisson bracket is not preserved $$ \tag{6} \{Q,P\}_{PB}~=~\lambda \{q,p\}_{PB}~\neq~\{q,p\}_{PB}~=~1. $$

Q2) Concerning OP's second question, it seems that OP is implicitly considering the third definition of a CT, i.e. a symplectomorphism. It is straightforward to see that a symplectomorphism $f:M\to M$ preserves the canonical volume form $$\tag{7} \Omega~:=~\frac{1}{n!}\omega^{\wedge n}$$ in phase space $(M,\omega)$, i.e. $$\tag{8} f^{\ast}\Omega=\Omega.$$

References:

  1. H. Goldstein, Classical Mechanics; Chapter 9. See text under eq. (9.11).

  2. L.D. Landau and E.M. Lifshitz, Mechanics; $\S45$. See text between eqs. (45.5-6).

  3. V.I. Arnold, Mathematical Methods of Classical Mechanics, 2nd eds., 1989; See $\S$44E and footnote 76 on p. 241.

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$^1$ Although Ref. 1 and Ref. 2 don't bother to mention this explicitly, it is implicitly assumed that the map (1) is a sufficiently smooth bijection, e.g., a diffeomorphism [which depends smoothly on the time parameter $t$]. Similar smoothness conditions are implicitly assumed about $H$, $K$, and $F$.

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