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What do expressions like

$$ \omega(X_f, X_g) $$

mean exactly?

(For context: I have, on the one hand, a chapter dealing with Hamiltonian IVPs as defined by Hamilton's equations $dp/dt = -\partial q/\partial t, dq/dt = -\partial p / \partial t$ and building up to the Poisson bracket and constants of motion; on the other, I have a chapter dealing with k-forms, symplectic structure, etc. building up to Darboux's theorem. But this expression

$$\{f,g\} = \omega(X_f, X_g) = dg(X_f) $$

I don't get. The third term is of course the Lie derivative, but what is $\omega$ doing there?)

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1 Answer 1

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The form $\omega$ is the symplectic form. It is a two-form so it can eat two vector fields and output a real function. The expression $X_f$ is presumably the Hamiltonian vector field associated to the function $f$. It is defined by the requirement that

$$ \omega(Y, X_f) = df(Y) $$

for all vector fields $Y$.

Finally, $\omega(X_f,X_g)$ is a real function obtained by plugging in two vector fields in a two-form and by the definition of the Hamiltonian vector field associated to $f$, we have

$$ \omega(X_f, X_g) = dg(X_f). $$

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