Friday, 10 January 2020

hamiltonian formalism - Poisson brackets' property as necessary and sufficient condition for transformation to be canonical?


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).



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 $$\tag{1} (q^i,p_i)~~\mapsto~~ \left(Q^i(q,p,t),P_i(q,p,t)\right)$$ [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.


No comments:

Post a Comment

Understanding Stagnation point in pitot fluid

What is stagnation point in fluid mechanics. At the open end of the pitot tube the velocity of the fluid becomes zero.But that should result...