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In this famous paper by Cary and Littlejohn on noncanonical Hamiltonian mechanics and its application to magnetic field line flow, they claim that as a result of Darboux's theorem, it is always possible to cast the vector potential one form $\mathbf{A}\cdot d\mathbf{x} = A_idx^i$ in canonical form $p\,dq - \mathcal{H}(p,q,t)\,dt$. I can see that this is possible by making a gauge choice that eliminates one of the terms from the vector potential one form, but how is this a corollary of Darboux's theorem?

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Given a $1$-form $$A~=~\sum_{I=1}^n A_I \mathrm{d}x^I \tag{1}$$ of constant rank in an odd-dimensional manifold $M$, it follows from the Darboux theorem that there exists a local coordinate system $$(\underbrace{q^1, \ldots, q^r}_{\text{positions}} ;\underbrace{p_1,\ldots, p_r}_{\text{momenta}}; \underbrace{c^1, \ldots, c^{n-1-2r}}_{\text{Casimirs}};\underbrace{t}_{\text{time}})\tag{2} $$ in a neighborhood $U\subseteq M$ such that the $1$-form $A$ is of the form $$A|_U ~=~ \sum_{i=1}^r p_i \mathrm{d}q^i -H \mathrm{d}t,\tag{3} $$ where $H$ is a constant (either $0$ or $-1$).

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