# Poisson brackets' property as necessary and sufficient condition for transformation to be canonical?

I read (Landau, Lifshitz: Mechanics) and then

I want to know if conditions (45.10) are sufficient for transformation $p,q \to P,Q$ to be canonical (obviously, they are necessary).

Condition (45.10) essentially defines a symplectomorphism. Some authors define a canonical transformation (CT) as a symplectomorphism, but not Landau & Lifshitz (L&L). They instead define a CT as a transformation $$\tag{1} (q^i,p_i)~~\mapsto~~ \left(Q^i(q,p,t),P_i(q,p,t)\right)$$ [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 $F$, see the text between eqs. (45.5-6).
Since a symplectomorhism (45.10) states nothing about $H$ and $K$, the condition (45.10) is not sufficient to be a CT according to L&L.
• As far as I know the two conditions are (locally) equivalent if L-L's condition is stated more precisely by adding some quantifiers. "For every $H$ there are $K$ and $F$ such that (2) holds." – Valter Moretti Nov 6 '17 at 10:45