Sign convention for Hamiltonian vector field.

Question

Let $(M,\omega)$ be a symplectic manifold and $f \in C^\infty(M)$. One can define the Hamiltonian vector field of $f$ as the only vector field ${\bf X}_f \in \mathfrak{X}(M)$ satisfying

(i) ${\rm d}f = \omega({\bf X}_f,\cdot)$ or

(ii) ${\rm d}f = \omega(\cdot, {\bf X}_f)$.

In other words, ${\rm d}f = \pm \iota_{{\bf X}_f}\omega$. What are the advantages and disadvantages of each sign convention? I'd think that (i) is more natural.

Answer 1

I think that a good answer for this is referring to what the book Introduction to symplectic topology of McDuff-Salamon comments about sign conventions. The whole point of the matter is that there is not only this sign convention: there is a whole range of places where we simultaneously want some sign or the other for psychological and/or practical reasons. Some examples being: the Hamilton equations, the standard complex structure, the standard symplectic form, the contraction, Floer's equation (imbued in this there is also the choice if we are taking the $-$gradient or gradient "flow"), the Poisson bracket and even the Lie bracket, to name a few. Juggling all this around and trying to keep practicality and psychological naturality of the definitions is hard.

The literal quote is:

There are various mutually inconsistent sign conventions in common use, and unfortunately it is impossible to make consistent choices yield all the identities which one would wish. We have chosen to arrange the signs so as to obtain the formula $[X_F,X_H]=X_{\{F,H\}}$ (...), and as a result a somewhat awkward minus sign appears somewhere else. Many authors deal with this by defining $X_H$ by $\iota(X_H)\omega_0=-dH$, but this gives the wrong sign for the Hamiltonian equations, unless the sign of the symplectic form $\omega_0$ is reversed as well, which is inconsistent with complex geometry. Another possibility is to reverse the sign in the definition of the Lie bracket for vector fields. This is contrary to costumary usage but is more natural.

The book says that Arnold's Mathematical methods in classical mechanics agrees with the sign choice of the Lie bracket (which I have the impression of being much more unusual than they imply, and personally not at all "natural"), but the sign choices conflict in other places.

So, the answer is that the advantages/disadvantages of a specific convention (for example, your case of the Hamiltonian vector field) are the entire cascade of what this convention implies for the other cases, and depends on your conventions in those other cases, which by themselves also depend on everything else etc. It is really more a web-like situation since these conventions are related to a plethora of rather basic objects of the theory, and not a linear strain of consequences of a convention for a single specific object (or a few objects at most). This is why I think sign conventions in symplectic geometry specifically, although abundant in mathematics as a whole, are more problematic and difficult for the literature to get in terms with.