# Which transformations are canonical?

1. Which transformations are canonical?

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

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 $F$.