I read (Landau, Lifshitz: Mechanics) and then
I want to know if conditions (45.10) are sufficient for transformation p,q→P,Q to be canonical (obviously, they are necessary).
Answer
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 (qi,pi) ↦ (Qi(q,p,t),Pi(q,p,t))
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.
Various definitions of CT and their interrelations are discussed in this Phys.SE post.
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