# Metriplectic Structure of a Radiation-Matter-Interaction Toy Model
## 5 The CMS in Madelung Variables
\begin{array}{rl}&\int^{\rho\sigma}(y)=\frac{\partial y^{\rho}}{ \partial x^{\mu}}(x(y))\frac{\partial y^{\sigma}}{\partial x^{\
u}}(x(y))J^ {\mu\
u}(x(y)),\\ &G^{\rho\sigma}(y)=\frac{\partial y^{\rho}}{\partial x^{\mu}}(x(y))\frac{ \partial y^{\sigma}}{\partial x^{\
u}}(x(y))G^{\mu\
u}(x(y))\end{array} \right. \tag{42}\]  
are achieved. Equation (42) state precisely that the Jacobi matrix \(J\) and the semimetric matrix \(G\) transform into the matrices \(J^{\prime}\) and \(G^{\prime}\), respectively, _as rank-2 tensors_ under any change of variables \(x\mapsto y(x)\). As the change of variables \((q,p,n)\mapsto(\phi,\rho,n)\) in (33) is applied to the calculation of \(f^{\rho\sigma}=\frac{\partial\varphi}{\partial x^{\mu}}\frac{\partial\varphi^{ \prime}}{\partial x^{\
u}}\int^{\mu\
u}\), the matrix \(J\) in (16) is transformed into the matrix \(J^{\prime}\) in (36), meaning that (33) is a _canonical change of variables_. In order to find the semimetric matrix \(G^{\prime}\) completing the CMS that represents the system (34), one applies the law \(G^{\prime\rho\sigma}=\frac{\partial\varphi}{\partial x^{\mu}}\frac{\partial \varphi^{\prime}}{\partial x^{\
u}}G^{\mu\
u}\) to the matrix \(G\stackrel{{\rm def}}{{=}}G_{\mathbb{E}}(q,p,n)\) in (24) with the gradients of (33), obtaining:  
\[\left\{ \begin{array}{l}G^{\prime\phi\phi}=2\partial_{q}\phi\partial_{p}\phi G ^{qp}+\left(\partial_{q}\phi\right)^{2}G^{qq}+\left(\partial_{p}\phi\right)^{ 2}G^{qp},\\ G^{\prime\phi\rho}=\partial_{q}q\partial_{q}G^{qq}+\partial_{p}\phi\partial_{ p}G^{pp}+\left(\partial_{q}\phi\partial_{p}\rho+\partial_{p}\phi\partial_{q} \phi\right)G^{qp},\\ G^{\prime\phi m}=\partial_{q}q\partial_{n}G^{qq}+\partial_{p}\phi\partial_{ n}nG^{mn},\\ G^{\prime p}=\left(\partial_{q}\rho\right)^{2}G^{qq}+\left(\partial_{p}\rho \right)^{2}G^{pp}+2\partial_{q}\rho\partial_{p}G^{qp},\\ G^{\prime\rho m}=\partial_{q}\rho\partial_{n}G^{qq}+\partial_{p}\rho\partial_{ n}nG^{pn},\\ G^{\prime mn}=(\partial_{n}n)^{2}G^{mn},\end{array} \right.\]  
that is  
\[\left\{ \begin{array}{l}G^{\prime\phi\phi}=0,\ \ G^{\prime\phi\rho}=0,\ \ G^{\prime\phi m }=0,\\ G^{\prime\varphi p}=\frac{8a^{2}n^{2}}{\chi G^{\prime}(n)},\\ G^{\prime\varphi n}=-\frac{2\alpha\
u\rho}{\chi G^{\prime}(n)},\\ G^{mn}=\frac{k\rho^{2}}{2\chi G^{\prime}(n)}.\end{array} \right. \tag{43}\]  
This is the matrix to be used in the composition of the symmetric bracket \(\chi(\cdot,S(n))\) to obtain the dissipative terms  
\[\eta_{\phi}=0,\ \ \eta_{\rho}=-2\alpha n\rho,\ \ \eta_{n}=\frac{k}{2}\rho^{2} \tag{44}\]  
to be added to the nondissipative ODEs (35) in order to obtain the dissipative ones (34). In particular, it is straightforward to check:  
\[\eta_{\phi}=\chi G^{\prime\phi m}S^{\prime}(n),\ \ \eta_{\rho}=\chi G^{\prime \rho n}S^{\prime}(n),\ \ \eta_{n}=\chi G^{mn}S^{\prime}(n), \tag{45}\]  
expressions to which \(\chi(\phi,S(n))\), \(\chi(\rho,S(n))\) and \(\chi(n,S(n))\) reduce, respectively. Last but not least, one observes that the Hamiltonian compatibility condition \(G^{\prime}\cdot\
abla^{\prime}H(\phi,\rho,n)=0\) is satisfied, considering the expression (38) for the Hamiltonian, as \(G^{\prime}\) in (43) is used, and \(\
abla^{\prime}\) is the gradient with respect to \(\phi\), \(\rho\) and \(n\). Wrapping up what has been found in the present Section, we may state that the TPA system can also be regarded as a CMS when the radiation is represented via the phase-amplitude variables \(\phi\) and \(\rho\), defined in (9), and that the new expression of the semimetric matrix \(G^{\prime}(\phi,\rho,n)\) is obtained by trasforming the matrix \(G_{\mathbb{E}}(q,p,n)\) as a rank-2 tensor under the transformations (33).