# Metriplectic Structure of a Radiation-Matter-Interaction Toy Model
## 4 Metriplectic Formulation
### The Metric Brackets Tensor  
Thanks to Equation (20), the system in Equation (18) becomes  
\[\left\{ \begin{array}{l}\dot{q}=\partial_{p}H+\chi G^{qp}S^{\prime}(n),\\ \dot{p}=-\partial_{q}H+\chi G^{pn}S^{\prime}(n),\\ \dot{n}=\chi G^{in}S^{\prime}(n),\end{array} \right. \tag{22}\]  
therefore, our overriding concern is to determine the tensor \(G\). In order to obtain such a result, we follow a linear algebraic procedure. Calculations are illustrated item-by-item in the Appendix A. The final result is:  
\[G_{\mathbb{E}}^{qq}=\frac{\partial_{q}H\partial_{n}H\Big{[}b\partial_{q}H \partial_{n}H+2c\partial_{p}H\sqrt{(\partial_{q}H)^{2}+(\partial_{p}H)^{2}+( \partial_{n}H)^{2}}\Big{]}}{[(\partial_{q}H)^{2}+(\partial_{p}H)^{2}]\,\big{[} (\partial_{q}H)^{2}+(\partial_{p}H)^{2}+(\partial_{n}H)^{2}\big{]}},\]  
\[G_{\mathbb{E}}^{qp}=\frac{\partial_{n}H\Big{[}b\partial_{q}H\partial_{p}H \partial_{n}H+c\big{[}(\partial_{p}H)^{2}-(\partial_{q}H)^{2}\big{]}\sqrt{( \partial_{q}H)^{2}+(\partial_{p}H)^{2}+(\partial_{n}H)^{2}}\Big{]}}{[( \partial_{q}H)^{2}+(\partial_{p}H)^{2}]\,\big{[}(\partial_{q}H)^{2}+(\partial _{p}H)^{2}+(\partial_{n}H)^{2}\big{]}},\]\[G_{\mathbb{E}}^{pn}=-\frac{b\partial_{q}H\partial_{n}H+c\partial_{q}H\sqrt{( \partial_{q}H)^{2}+(\partial_{p}H)^{2}+(\partial_{n}H)^{2}}}{(\partial_{q}H)^{2 }+(\partial_{p}H)^{2}+(\partial_{n}H)^{2}},\]  
\[G_{\mathbb{E}}^{pn}=\frac{b(\partial_{p}H)^{2}(\partial_{n}H)^{2}-2c\partial_{ q}H\partial_{p}H\partial_{n}H\sqrt{(\partial_{q}H)^{2}+(\partial_{p}H)^{2}+( \partial_{n}H)^{2}}}{[(\partial_{q}H)^{2}+(\partial_{p}H)^{2}]\left[(\partial _{q}H)^{2}+(\partial_{p}H)^{2}+(\partial_{n}H)^{2}\right]},\]  
\[G_{\mathbb{E}}^{pn}=\frac{-b\partial_{p}H\partial_{n}H+c\partial_{q}H\sqrt{( \partial_{q}H)^{2}+(\partial_{p}H)^{2}+(\partial_{n}H)^{2}}}{(\partial_{q}H)^ {2}+(\partial_{p}H)^{2}+(\partial_{n}H)^{2}},\]  
\[G_{\mathbb{E}}^{pn}=\frac{b\left[(\partial_{q}H)^{2}+(\partial_{p}H)^{2} \right]}{(\partial_{q}H)^{2}+(\partial_{p}H)^{2}+(\partial_{n}H)^{2}},\]  
with  
\[\left\{ \begin{array}{l}b=\frac{\Psi_{g}}{\chi S^{(n)}}\frac{(\partial_{q}H) ^{2}+(\partial_{p}H)^{2}+(\partial_{n}H)^{2}}{(\partial_{q}H)^{2}+(\partial_{ p}H)^{2}},\\ c=\frac{\Psi_{p}\partial_{q}H-\Psi_{q}\partial_{p}H}{\chi S^{(n)}}\frac{ \sqrt{(\partial_{q}H)^{2}+(\partial_{p}H)^{2}+(\partial_{n}H)^{2}}}{(\partial _{q}H)^{2}+(\partial_{p}H)^{2}}.\end{array} \right. \tag{23}\]  
In the system at hand, due to the nature of the dissipation terms \(\psi_{q}\) and \(\psi_{p}\) in (8) and (15), it is easy to see that \(\psi_{q}\partial_{p}H-\psi_{p}\partial_{q}H=0\), so that the term \(c\) in (23) simply vanishes. This is definitely not a general condition, rather it requires the two variables \(q\) and \(p\) to be dissipated formally in the same way, and the Hamiltonian \(H\) to be symmetric under the exchange of \(q\) and \(p\). A simple counterexample for which \(c\
eq 0\) is illustrated in the Appendix, with the result (A7). The set of components of the metric tensor \(G_{\mathbb{E}}\) may be finally expressed in a much more compact, and yet explicit form: putting together the expression \(H(q,p,n)\), the values of \(b\) in (23), and \(c=0\), after some algebra, the expressions  
\[\left\{ \begin{array}{l} G_{\mathbb{E}}^{qq}=\frac{8a^{2}}{\chi S ^{\prime}(n)}\frac{q^{2}n^{2}}{\left(q^{2}+p^{2}\right)^{2}},\ \ G_{\mathbb{E}}^{qp}=\frac{8a^{2}}{\chi S ^{\prime}(n)}\frac{qpn^{2}}{\left(q^{2}+p^{2}\right)^{2}},\ \ G_{\mathbb{E}}^{ pp}=\frac{8a^{2}}{\chi S^{\prime}(n)}\frac{p^{2}n^{2}}{\left(q^{2}+p^{2} \right)^{2}},\\ G_{\mathbb{E}}^{pn}=-\alpha\frac{nq}{\chi S^{\prime}(n)},\ \ G_{\mathbb{E}}^{ pn}=-a\frac{np}{\chi S^{\prime}(n)},\ \ G_{\mathbb{E}}^{pn}=\frac{k}{8\chi S^{\prime}(n)} \left(q^{2}+p^{2}\right)^{2}\\ \text{are written}.\end{array} \right.