This form of the action slightly differs from that proposed in Ref. [14]. The main difference consists in introducing the vector potential for the magnetic field. Therefore, here the canonical pair is $\mathbf{A}$, $-\mathbf{M}$ instead of $\mathbf{H}$, $\mathbf{S}$, where $\mathbf{S}=\text{curl}\mathbf{M}$. We do not consider the discontinuous flows and thus we omit the surface term in the action. But adding corresponding surface term we can easily take the breaks into account.

Note that substituting $\mathbf{M}=\text{curl}\mathbf{S}$ into Eq. (11) one can integrate it and arrive at the dynamic equation for $\mathbf{S},{\partial }_t\mathbf{S}={\left(4\pi \right)}^{-1}\mathbf{H}+\mathbf{v}\times \text{curl}\mathbf{S}+\nabla \Psi$, where $\Psi$ represents a scalar field respectful for the $\mathbf{S}$ gauge. This relation differs only by the $\mathbf{S}$ sign from Eq. (10.9) of reference [1] [or Eq. (7) in the original paper [22]].

We do not include $\Lambda$ into the set of canonical variables dealing with the extended Hamiltonian description, see Refs. [30, 31]. Otherwise, we can include $\Lambda$ into the set of generalized coordinates. Denoting corresponding conjugate momentum ${\pi }_{\Lambda }$ and adding to the Hamiltonian density, Eq. (18), the term ${\pi }_{\Lambda }\nu$ results in$${\partial }_t{\pi }_{\Lambda }=-\delta \mathcal{H}∕\delta \Lambda =-\text{div}\mathbf{M},{\partial }_t\Lambda =\delta \mathcal{H}∕\delta {\pi }_{\Lambda }=\nu .$$Variation of the action with respect to the additional variable $\nu$ results now in the restriction ${\pi }_{\Lambda }=0$. This restriction is consistent with the set of canonical equations. Namely, Eq. (11) (or, equivalently, the canonical equation ${\partial }_t\mathbf{M}=\delta H∕\delta \mathbf{A}$) leads to ${\partial }_t\text{div}\mathbf{M}=0$. The momentum ${\pi }_{\Lambda }$ is the linear function of $t,{\pi }_{\Lambda }={\pi }_{\Lambda }\left(t_0\right)-\left(t-t_0\right)\text{div}\mathbf{M}$. The condition ${\pi }_{\Lambda }=0$ follows for the specific choice ${\pi }_{\Lambda }\left(t_0\right)=0$ and $\text{div}\mathbf{M}\left(t_0\right)=0$. Moreover, as it becomes clear below, the initial condition $\mathbf{M}\left(t_0\right)=0$ [resulting in $\text{div}\mathbf{M}\left(t_0\right)=\text{div}\mathbf{M}\left(t\right)=0$] leads to essential simplifications. On the other hand, conditions ${\pi }_{\Lambda }=0,\mathbf{M}\left(t_0\right)=0$ do not lead to any restrictions on the physical variables and the MHD flow as well. The subsidiary functions $\Lambda$ and $\nu$ can be expressed in terms of other variables as $\Lambda ={\Delta }^{-1}\left(\text{div}\left[\mathbf{v}\times \mathbf{H}\right]-\text{div}\dot{\mathbf{A}}\right)+{\Lambda }^{\prime },\nu ={\partial }_t\Lambda$, where $\Delta$ denotes the Laplace operator and ${\Lambda }^{\prime }$ is arbitary solution of the Laplace equation.

Note that the considerations based upon the Pfaff’s theorem also result in the reduced velocity representation. But this theorm in our case claims only the local equivalence between the three-dimensional vector field and the standard form $\varphi +\lambda \nabla \mu$ with the appropriate scalars $\varphi ,\lambda ,\mu$. This point is often ignored.

In terms of the differential forms $\alpha$ and $\mathbf{I}$ are scalar and vector $0$ forms; $\mathbf{L},\mathbf{J}$, and $\rho$ and one-, two-, and three-forms, respectively.