# Metriplectic Structure of a Radiation-Matter-Interaction Toy Model
## 5 The CMS in Madelung Variables  
In this Section, we are going to present the same CMS as before, but describing the electromagnetic phasor \(\psi\) via its square-root amplitude and phase variables \(\rho\) and \(\phi\) as defined in (9), instead of the two quantities \(q\) and \(p\). The variables \((\phi,\rho)\) are related to the \((q,p)\) ones via the transformation:  
\[\left\{ \begin{array}{rl}q=\sqrt{2\rho}\cos\phi,&\left\{\begin{array}{rl} \phi=-\arctan\Big{(}\frac{p}{q}\Big{)},\\ p=-\sqrt{2\rho}\sin\phi,&\left\{\begin{array}{rl}\rho=\frac{1}{2}(q^{2}+p^ {2}).\end{array}\right.\end{array}\right.\end{array} \right. \tag{33}\]  
These are invertible and smooth, except for \(\rho=0\), that is, the singular point at which no radiation exists at all. It is easy to see that the new set of variables \((\phi,\rho,n)\) evolves according to the ODEs  
\[\left\{ \begin{array}{rl}\phi=\rho,&\\ \dot{\rho}=-2\alpha n\rho,&\\ \dot{n}=\frac{k}{2}\rho^{2},&\end{array} \right. \tag{34}\]as the system \((q,p,n)\) undergoes (8). The \(\alpha\to 0\) and \(k\to 0\) limit of (34) gives the non-dissipative approximation of those equations:  
\[\left\{ \begin{array}{rl}\dot{\phi}=\rho,\ \ \dot{\rho}=0,\\ \dot{n}=0.\end{array} \right. \tag{35}\]  
Equation (35) suggest the angle-frequency nature of the variables \(\phi\) and \(\rho\) in the Hamiltonian limit. According to them, \(\rho\) is a constant of motion, while \(\phi(t)=\rho t+\phi(0)\) evolves linearly with time. Dissipation, as in (34), slows down the run of \(\phi(t)\) as it consumes \(\rho\) in favour of the population \(n\). Equations (35) form a Hamiltonian system, provided one uses the symplectic tensor  
\[J^{\prime}(\phi,\rho,n)=\left( \begin{array}{ccc}0&1&0\\ -1&0&0\\ 0&0&0\end{array} \right) \tag{36}\]  
and the Hamiltonian  
\[H_{0}(\rho)=\frac{1}{2}\rho^{2}. \tag{37}\]  
This \(H_{0}(\rho)\) is not conserved under the motion (34): the extension  
\[H(\phi,\rho,n)=\frac{1}{2}\rho^{2}+\frac{2\alpha}{k}n^{2}+U_{0} \tag{38}\]  
of (37) closes the system energetically, because \(H(\phi,\rho,n)\) is conserved under the motion (34) (moreover, this \(H\) does not truely depend on \(\phi\)). Nonetheless, it is not possible to generate the whole dynamics (34) via just the Hamiltonian (38) and the PB given by (36): the system (34) can be rather represented as a _complete metriplectic system_ by adding to \(\{\cdot,H(\phi,\rho,n)\}\) a metric bracket expressed in terms of \(\phi\), \(\rho\) and \(n\). In order to complete the system (34) as a CMS, one may either re-formulate ex novo the whole problem, as achieved in the \((q,p,n)\) variables, repeating everything in the new \((\phi,\rho,n)\), through the Equation (34), the Hamiltonian (38) and the PB given by the Jacobi tensor (36); or, as will be the case here, by appreciating the _tensor nature_ of the metriplectic laws [6]  
\[\dot{x}^{\mu}=J^{\mu\
u}(x)\partial_{v}H(x)+\chi G^{\mu\
u}(x)\partial_{v}S(x) \tag{39}\]  
under any diffeomorphic change of variables \(x\mapsto y(x)\). Indeed, provided \(H\) and \(S\) are _scalar quantities_ under the transformations \(x\mapsto y(x)\), one may perform the variable change, and obtain:  
\[\dot{y}^{\rho}=\frac{\partial y^{\rho}}{\partial x^{\mu}}\frac{\partial y^{ \sigma}}{\partial x^{\
u}}J^{\mu\
u}(x(y))\frac{\partial H(y)}{\partial y^{ \sigma}}+\chi\frac{\partial y^{\rho}}{\partial x^{\mu}}\frac{\partial y^{ \sigma}}{\partial x^{\
u}}G^{\mu\
u}(x)\frac{\partial S(y)}{\partial y^{ \sigma}}. \tag{40}\]  
This (40) is equivalent to the initial system (39), and it may be put into an explicitly CMS-like form  
\[\dot{y}^{\rho}=J^{\prime\rho\sigma}(y)\frac{\partial H(y)}{\partial y^{ \sigma}}+\chi G^{\rho\sigma}(x)\frac{\partial S(y)}{\partial y^{\sigma}}, \tag{41}\]  
provided the identifications  
\[\left\{