# System with non-skew symmetric gyroscopic matrix

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

$$T=\frac{1}{2}\left(m\dot{u}_{s}^{2}+L\dot{q}^{2}+\varphi'u\dot{q}\right)$$

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

$$\mathscr{L}=T-V=\frac{1}{2}\left(m\dot{u}_{s}^{2}+L\dot{q}^{2}+\varphi'u_{s}\dot{q}\right)-\frac{1}{2}\left(ku_{s}^{2}+C^{-1}q^{2}\right)$$

we obtain

$$\frac{1}{2}m\dot{u}_{s}^{2}+\frac{1}{2}L\dot{q}^{2}-V=const$$

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?