Lets take the following equations of motions of a two DOF system

$$\left[\begin{array}{cc} m & 0\\ 0 & L \end{array}\right]\left[\begin{array}{c} \ddot{u}\\ \ddot{q} \end{array}\right]=-\left[\begin{array}{cc} 0 & 0\\ \varphi' & 0 \end{array}\right]\left[\begin{array}{c} \dot{u}\\ \dot{q} \end{array}\right]-\left[\begin{array}{cc} k & 0\\ 0 & C^{-1} \end{array}\right]\left[\begin{array}{c} u\\ q \end{array}\right]+\left[\begin{array}{c} F_{m}\\ F_{e} \end{array}\right]$$

where $u$ is the position, q the charge, $m$ the mass, $L$ the inductance, $k$ the mechanical stiffness, $C$ the capacitance and $\varphi'$ the magnetic flux derivative with respect to the unique direction $x$. $F$ represents external forces.

I derived the the kinetic energy of this system as


which, for me, makes no sense since the third term in the RHS is not symmetric; i.e. it has not a $-\varphi'q\dot{u}$ couterpart. If it had, then I could understand the system as a non-natural gyroscopic system.

My intuition says that there is something wrong with this system. Following it, I tried a justification stating that for a non-natural conservative system the Jacobi's integral gives $$\sum\frac{\partial\mathscr{L}}{\partial\dot{q}}\dot{q}-\mathscr{L}=const$$

Being the Lagrangian of the system


we obtain


Then, recalling the above expression of the kinetic energy, we can say that the system does not conserves energy.

My question is, is that reasoning correct? Can I say this system has the wrong physics?


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