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I read (Landau, Lifshitz: Mechanics) enter image description here and then enter image description here


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

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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.

Various definitions of CT and their interrelations are discussed in this Phys.SE post.

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    $\begingroup$ 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." $\endgroup$ – Valter Moretti Nov 6 '17 at 10:45
  • $\begingroup$ @Valter Moretti: Agree. $\endgroup$ – Qmechanic Nov 6 '17 at 11:24
  • $\begingroup$ Sorry for the late reply, What does it mean that two conditions are equivalent locally? $\endgroup$ – LRDPRDX Jun 7 '18 at 10:17
  • $\begingroup$ The word locally means in this context in a sufficiently small open neighborhood. $\endgroup$ – Qmechanic Jun 7 '18 at 10:25

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