TlDr; a different choice of symplectic structure on a phase-space $\mathcal{M}$ affects the Hamiltonian mechanics insofar as it could affect what the canonical coordinates are, but is this the only noteworthy difference?
I've recently been doing some novice-level reading into classical mechanics (for the mathematically inclined), via symplectic manifolds (main source has been: "Introduction to mechanics and symmetry", by Marsden and Ratiu). I've come up with the following vague picture of the physical relevance of the symplectic form (please bear with me, or skip, this is just context for the main question which is at the bottom of the post):
- We want smooth functions (observables) $F:\mathcal{M}\to\Bbb R$ (i.e. $F\in\Omega^0(\mathcal{M})$ is a zero-form) to induce vector fields on $\mathcal{M}$, thought of as time-evolution flows associated to the observable which make $F$ constant
- Since making the observable $F$ constant asks only about how it changes, when it comes to figuring out this "induced vector field" formalism we only care about the $1$-form $dF\in\Omega^1(\mathcal{M})$
- We can't a priori ask for much control over what $dF$ looks like, so we should examine the general problem of associating $1$-forms with vector fields; thus, we decide we want some kind of (physically meaningful?) isomorphism $\mathscr{T}(\mathcal{M})\cong\mathscr{T}^\ast(\mathcal{M})$, at least in the finite-dimensional case
- Such an isomorphism always exists (we can pick a metric) but we want to do better than that. It really boils down to deciding on a good notion of pairing: $\mathscr{T}(\mathcal{M})\otimes\mathscr{T}^\ast(\mathcal{M})\to\mathcal{M}\times\Bbb R$ (I am using bundle, rather than sheaf, notation). There are some options; we obviously want it to be (bi)linear, but we could chuck in some adjectives such as symmetric (almost the definition of a Riemannian metric) or alternating (giving a $2$-form) or perhaps some other thing
- But we notice, since we want $dF(X_F)=0$, it should hold that (if $\langle X_F,-\rangle$ is the $1$-form associated with a vector field suggestively called $X_F$ via our pairing, which we have yet to define) $\langle X_F,X_F\rangle=0$ which throws the metric idea out the window and exactly says we want to be alternating
- So, we are forced to consider $2$-forms $\omega\in\Omega^2(\mathcal{M})$ to solve our problem of how to formalise this 'induced flow' business. I've heard that in some physical problems, we want $\omega$ to be closed, sometimes not, but certainly the condition that $\omega$'s associated pairing makes $\mathscr{T}(\mathcal{M})\cong\mathscr{T}^\ast(\mathcal{M})$ means it should be symplectic (maybe modulo $d\omega\overset{?}{=}0$, but energy conservation ideas lead to wanting $d\omega=0$ in most cases, I think) so - great - we are forced to consider symplectic forms
- We also learn that symplectic forms are locally trivialisable; if $\dim\mathcal{M}=2n$ then every point has an open neighbourhood $U$ and diffeomorphism $U\cong\Bbb R^{2n}$ in which $\omega$ is associated with $\sum_i dp_i\wedge dq_i$, where $(q_1,q_2,q_3,\cdots,q_n,p_1,p_2,\cdots,p_n)$ is our diffeomorphism.
So far, the only interesting observation seems to be the final bullet point. Taking some hints which I've seen in various places, I'm interpreting the final bullet point as saying something like this:
The symplectic form gives a notion of canonical coordinates, and in particular if $\mathcal{M}$ is associated with a configuration space $\mathcal{Q}$ as $\mathscr{T}^\ast(\mathcal{Q})$, or something similar, it allows coordinates on $\mathcal{Q}$ to dictate the "conjugate momenta" coordinates, the $p_i$ associated to every given $q_i$.
As a novice, I'm not yet entirely convinced of how physically important conjugate momenta are, but I can already see it's an important concept in connecting Lagrangian and Hamiltonian mechanics and it seems to be useful for choosing convenient coordinates in which a problem simplifies (please, add to this if you can).
Main question:
Is the determination of conjugate momenta the only (or only especially important) effect a choice of symplectic form has on the resulting mechanics?
If $\omega$ is symplectic, so is $2\omega$. If $F$ is an observable, and $X_F$ and $X_F'$ are the vector fields induced by $\omega,2\omega$ respectively, then $X_F'=2X_F$ and I imagine the time evolution (the trajectories of the particle, the answer to the actual physical problem) is just affected by a simple respeeding, or we could change our physical units (coordinate functions) by a rescaling and keep the time units the same; the conjugate momenta should become simply doubled or halved, but this is unimportant (right?). Now, of course, there could be more complicated ways of replacing a given symplectic form with another one, e.g. by considering $\omega'=f\wedge\omega$ for an always-positive $0$-form $f$, but to what extent does the physicist care? Is the goal, when expressing a problem in the Hamiltonian formalism, just to find a symplectic structure and then profit? Or does the physicist think deeply about the choice of structure?
Of course, canonical ones can be easy to write down and arise directly from prior experience, e.g. I can see why we always use the canonical form on $\mathscr{T}^\ast(\mathcal{Q})$. But, since in local coordinates all equations can be made to look the same, I do wonder how important this really is (e.g. we always get Hamilton's equations, it's just that the conjugate $p$ to a given $q$ might be different). I'm particularly interested in an answer to this question because I've also been learning some quantum theory, and heard that we can put a symplectic structure on phase Hilbert space $\mathbb{P}(\mathcal{H})$ but this structure strikes me as unmotivated; if it turns out the particular choice doesn't affect the physics very much, I could sleep (study) more easily.
