Which transformations are canonical?

Question

Which transformations are canonical?

Answer 1

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

2. Secondly, Wikipedia (October 2015) calls a transformation (1) [together with a choice of $H(q,p,t)$ and $K(Q,P,t)$] a CT if it transforms the Hamilton's eqs. into Kamilton's eqs. This is called a canonoid transformation in Ref. 3.

3. Thirdly, Ref. 3 calls a transformation (1) a CT if $\forall H(q,p,t) \exists K(Q,P,t)$ such that the transformation (1) transforms the Hamilton's eqs. into Kamilton's eqs.

4. Fourthly, some authors (e.g. Ref. 4) 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. $$ f^{\ast}\omega=\omega.\tag{3}$$ Here $\omega$ is the symplectic two-form, which in local Darboux/canonical coordinates reads $\omega= \sum_{i=1}^n\mathrm{d}p_i\wedge \mathrm{d}q^i$. A symplectomorphism preserves the Poisson bracket.

5. Fifthly, Ref. 1 defines an extended canonical transformation (ECT) as a transformation (1) [together with a choice of $H(q,p,t)$ and $K(Q,P,t)$] that satisfies $$ \lambda(\sum_{i=1}^np_i\mathrm{d}q^i-H\mathrm{d}t) -(\sum_{i=1}^nP_i\mathrm{d}Q^i -K\mathrm{d}t) ~=~\mathrm{d}F \tag{4}$$ for some parameter $\lambda\neq 0$ and for some generating function $F$.

Now let us discuss some of the relationships between the above five 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 implies $$ \sum_{i=1}^n\mathrm{d}p_i\wedge \mathrm{d}q^i-\mathrm{d}H\wedge \mathrm{d}t ~=~ \sum_{i=1}^n\mathrm{d}P_i \wedge \mathrm{d}Q^i -\mathrm{d}K\wedge \mathrm{d}t,\tag{5} $$ and is therefore 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: $$ Q~=~\lambda q, \qquad P=p \qquad K~=~\lambda H, \qquad F~=~0,\tag{6}$$ where $\lambda\notin \{0,1\}$ is a constant different from zero and one, so that the Poisson bracket is not preserved $$ \{Q,P\}_{PB}~=~\lambda \{q,p\}_{PB}~\neq~\{q,p\}_{PB}~=~1. \tag{7} $$

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. J.V. Jose & E.J. Saletan, Classical Dynamics: A Contemporary Approach, 1998; Subsection 5.3.1, p. 233.

4. 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 $F$.