classical mechanics - Which transformations are canonical?

Which transformations are canonical?

Answer

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 $\omega= \sum_{i=1}^n\mathrm{d}p_i\wedge \mathrm{d}q^i$.
   
   


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 $F$.