# Which transformations are canonical?

Which transformations are canonical?

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

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 $$F$$.